Bounds for graph regularity and removal lemmas

Bounds for graph regularit y and remo v al lemmas Da vid Conlon ∗ Jacob F o x † Abstract W e show, for a ny positive integer k , that there exis ts a graph in which a ny equitable par tition of its vertices into k parts has at lea st ck 2 / log ∗ k pairs of parts which a re no t ǫ -regular, wher e c, ǫ > 0 are absolute c onstants. This b ound is tight up to the constant c and addresses a questio n of Gow ers on the num ber of ir r egular pairs in Szemer´ edi’s regularity lemma. In order to gain some control o v er irregular pairs, another reg ularity lemma, known a s the strong regular ity lemma, w as develop ed by Alon, Fisc her, Krivelevich, a nd Szegedy . F or this lemma, we prov e a low er b o und of wo wzer-type, whic h is one level higher in the Ac k ermann hierarch y than the tow er function, on the num ber of parts in the strong regularity lemma, esse ntially matc hing the upper b ound. On the other hand, for the induced graph r emov al lemma, the s ta ndard applicatio n of the str o ng regularity lemma , we ﬁnd a diﬀerent pro of which yields a tow er-t yp e b ound. W e also discuss b ounds on several related regularity lemmas, including the weak regularity lemma of F rieze a nd Kanna n and the recently established reg ular approximation theorem. In particular, we show that a weak partition with approximation parameter ǫ may re q uire as man y as 2 Ω( ǫ − 2 ) parts. This is tight up to the implied constant and so lves a problem studied by Lov´ asz and Szegedy . 1 In tro duction Originally deve lop ed by Szemer ´ edi as part of his pr o of of the celebrated Erd ˝ os-T ur´ an conjecture on long arithmetic progressions in dense subsets of th e intege rs [39], Szemer ´ edi’s regularit y lemma [40] h as b ecome a cen tral tool in extremal com binatorics. Roughly sp eaking, the lemma sa ys that the v erte x set of any graph ma y b e partitioned in to a small n umber of parts suc h that the b ipartite subgraph b et w een almost ev ery pair of parts b eha v es in a random-lik e fashion. Giv en t w o subsets X and Y of a graph G , w e write d ( X, Y ) for the densit y of edges b et w een X and Y . The pair ( X, Y ) is said to b e ( ǫ, δ ) -r e gular if f or some α an d all X ′ ⊂ X and Y ′ ⊂ Y with | X ′ | ≥ δ | X | and | Y ′ | ≥ δ | Y | , we h a v e α < d ( X ′ , Y ′ ) < α + ǫ . In the case where δ = ǫ , w e say that the pair ( X , Y ) is ǫ -r e gular . By sa ying that a pair of parts is r an d om-lik e, w e mean that th ey are ( ǫ, δ )-regular with ǫ and δ sm all, a pr op ert y whic h is easily seen to b e satisﬁed with h igh p robabilit y by a random bipartite graph. W e will also ask that the diﬀerent parts b e of comparable size, that is, that the partition V ( G ) = V 1 ∪ . . . ∪ V k b e e quitable , that is, || V i | − | V j || ≤ 1 for all i and j . ∗ St John’s College, Cambridge CB2 1TP , United Kingdom. E-mail: d.conl on@dpmms.cam .ac.uk . Researc h supp orted by a Roya l So ciety Universit y R esearc h F el lo wship. † Department of Ma thematics, MIT, Cambridge, MA 02139-430 7. E-mail: fox@math.mit.edu . Researc h sup p orted by a Simons F ellowship and NSF gran t D MS-1069197. 1 The r e gularity lemma no w states that f or eac h ǫ, δ , η > 0, there is a p ositiv e integ er M = M ( ǫ, δ , η ) suc h that the vertice s of an y graph G can b e equitably partitioned V ( G ) = V 1 ∪ . . . ∪ V M in to M parts where all but at most an η fraction of the p airs ( V i , V j ) are ( ǫ, δ )-regular. W e shall sa y that su c h a partition is ( ǫ, δ, η )-regular and simp ly ǫ -regular in the case ǫ = δ = η . F or more bac kground on the regularit y lemma, see the excellen t sur v eys by Koml´ os and Simonovits [27] and R¨ odl and Sc h ac ht [33]. Use of the regularit y lemma is no w widespread throughout graph theory . Ho we v er, one of the earliest applications, the triangle remov al lemma of Ru zsa and Szemer´ edi [36], remains the standard example. It states that for an y ǫ > 0 there exists δ > 0 suc h that any graph on n v ertices w ith at m ost δ n 3 triangles can b e made triangle-free by remo ving ǫn 2 edges. It easily implies Roth’s theorem [34] on 3- term arith m etic progressions in dense sets of in tegers, and Solymosi [37] sh o wed that it furth er implies the str on ger corners theo rem of Ajtai and Szemer ´ edi [1], which states that an y dense subset of the in teger grid con tains the v ertices of an axis-alig ned isosceles triangle . T his resu lt w as extended to all graphs in [17, 3]. The extension, kn own as the graph remo v al lemma, sa ys that given a graph H on h v ertices and ǫ > 0 there exists δ > 0 suc h that an y graph on n vertice s with at most δ n h copies of H can b e made H -fr ee b y remo ving ǫn 2 edges. One disadv antag e of applying the regularit y lemma to prov e this theorem is the b ounds that it giv es for the size of δ in terms of ǫ . The pro of of the regularit y lemma yields a b ound of to we r-t y p e for the num b er of piece s in the p artition. When this is applied to graph remo v al, it giv es a b ou n d for δ − 1 whic h is a to wer of tw os of heigh t polynomial in ǫ − 1 . S urpr isin gly , any hop e that a b etter b ound for the regularit y lemma might b e found w as pu t to rest b y Go wers [22], who show ed that there are graphs for whic h a to wer-t yp e n umber of parts are r equired in ord er to obtain a regular p artition. T o b e more pr ecise, the p ro of of the regularit y lemma sho w s that M ( ǫ, δ, η ) can b e tak en to b e a to w er of t w os of heigh t prop ortional to ǫ − 2 δ − 2 η − 1 . Go wers’ result, d escrib ed in [13] as a tour de force, is a lo wer b oun d, with c = 1 / 16, on M (1 − δ c , δ, 1 − δ c ) whic h is a tow er o f tw os o f h eight prop ortional to δ − c . As Go w ers notes, it is an easy exercise to translate lo we r b ounds for small δ an d large ǫ into lo wer b ound s for large δ and small ǫ which are also of to w er-t yp e. Ho w ev er, the natural question, d iscussed b y Szemer ´ edi [40], Koml´ os and Simonovits [27], and Go w ers [22], of d etermin in g the dep enden cy of M ( ǫ, δ , η ) on η , w hic h measures the fr action of allo wed irregular pairs, has remained op en. Th is is the ﬁrst p roblem we will address here, sh o win g that the d ep endence is aga in of to wer-t yp e. This do es n ot mean that b etter b ounds cannot b e pr o ved for the graph remo v al le mma. Recen tly , an alternativ e proof was found b y the second author [18], allo wing one to sh o w that δ − 1 ma y b e tak en to b e a to wer of tw os of h eigh t O (log ǫ − 1 ), b etter than one could p ossibly d o using regularit y . T hough this remains qu ite far from the lo wer boun d of ǫ − O (log ǫ − 1 ) , it clears a signiﬁcan t h u rdle. The second ma jor theme of this pap er is a p ro of of the induced graph r emo v al lemma which similarly b ypasses a natural obstacl e. Let H b e a graph on h ve rtices. The induced graph remo v al lemma, pro v ed b y Alon, Fisc h er, Krivelevi c h , and Szegedy [5], states that for any ǫ > 0 there exists δ > 0 su ch that an y graph on n vertice s with at most δ n h induced copies of H m a y b e made induced H -fr ee b y adding or deleting at most ǫn 2 edges. This result, wh ich easily imp lies the graph remov al lemma, do es not readily follo w from the same 2 tec hniqu e used to pr o ve the graph r emov al lemma, mainly b ecause of the p ossib ility of irr egular pairs in the r egularit y p artition. T o o v ercome th is issue, Alon, Fisc h er, K riv elevic h, and Szegedy [5 ] dev elop ed a strenghthening of Szemer ´ edi’s regularit y lemma. Roughly , it giv es an equitable partition A and an equitable reﬁ nemen t B of A suc h that A and B are b oth regular, with the guarante ed regularit y of B allo wed to d ep end on the size of A , and the edge density b et ween almost all pairs of parts in B close to the edge density b et ween the pair of parts in A that they lie in. The pro of of the stron g regularit y lemma inv olv es iterativ e applications of Szemer´ edi’s regularit y lemma. Th is causes the upp er b ound on the n umb er of parts in B to gro w as a w o wzer fu nction, whic h is one lev el higher in the Ac k erm ann hierarc hy than the to w er fun ction. In order to get an impro v ed b ound for its v arious applications, one ma y hop e that an impro v ed b ound of to wer-t yp e could b e established. W e sh o w that no suc h b ound exists. In fact , w e will show that a seemingly w eak er s tatemen t requires w o w zer-t yp e b ound s. On the other h and, w e give an alternativ e pro of of the indu ced graph remov al lemma, allo win g one to sh o w that δ − 1 ma y b e take n to b e a tow er o f t w os of heigh t p olynomial in ǫ − 1 , b etter than one could p ossibly ac hiev e usin g the strong r egularit y lemma. W e also make p rogress on determinin g b ou n ds for v arious related regularity lemmas, including the F rieze-Ka nnan weak regula rit y lemma [19, 20] and the regular appro ximation theorem, due indep en- den tly to Lo v´ asz a nd S zegedy [29] and to R¨ odl and Sc hac ht [32]. W e discu s s all these con trib utions in more d etail in the sectio ns b elo w. 1.1 The num b er of irr egular pairs The role of η in the regularit y lemma is to measure how many p airs of subsets in the p artition are regular. If a partition in to k pieces is ( ǫ, δ, η )-regular, then there will b e at most η  k 2  irregular pairs in the partition. S zemer ´ edi [40] wrote that it would b e int eresting to determine if th e assertion o f the regularit y lemma holds w hen w e do not allo w any irregular p airs. This question remained unan s w er ed for a long time u n til it was observ ed by Lo v´ asz, Seymour, T rotter, and Alon, Duke , Lefmann, R¨ odl, and Y uster [3] that irregular pairs can b e necessary . Th e simple example of the half-graph sh o ws that this is indeed the case. The half-graph is a bipartite graph with verte x sets A = { a 1 , . . . , a n } and B = { b 1 , . . . , b n } in whic h ( a i , b j ) is an edge if and only if i ≤ j . Any partitio n of this graph in to M parts will ha ve Ω( M ) irregular pairs. In other w ords, M ( ǫ, δ, η ) m ust gro w at least linearly in η − 1 . Ho wev er, the n umb er of irregular pairs, or, in other wo rds, the dep endence of M ( ǫ, δ, η ) on η − 1 with ǫ and δ ﬁxed, h as not b een w ell unders too d despite b eing asked sev eral times, in cluding by Koml´ os and Sim on ovits [27] and , more explicitly , b y Go wers [22 ]. This problem and r elated problems ha ve con tinued to attract in terest (see, e.g., [26], [30]). The linear b oun d obtained from the half-graph app ears to b e the only b ound in the literature for th is p r oblem. F or ﬁxed constants ǫ and δ and eac h M , we giv e a construction in whic h an y partition in to M parts has at least cM 2 / log ∗ M irregular p arts, where c > 0 is an absolute co nstan t, and this is tigh t apart from the constant c . The iterated logarithm log ∗ n is the n um b er of times the logarithm fu nction needs to b e applied to get a n umber w h ic h is at most 1. That is, log ∗ x = 0 if x ≤ 1 and otherwise log ∗ x = 1 + log ∗ (log x ) denotes the iterated logarithm. In other w ords, the dep end ence on η in 3 M ( ǫ, δ , η ) is indeed a to wer of t wos of heigh t p rop ortional to Θ( η − 1 ). Theorem 1.1 Ther e ar e absolute c onstants c, ǫ, δ > 0 su c h th at for e very k ther e is a gr aph in which every e quitable p artition of the gr aph into k p arts has at le ast ck 2 / log ∗ k p airs of p arts which ar e not ( ǫ, δ ) -r e gular. In other wor ds, M ( ǫ, δ , η ) is at le ast a tower of two s of height cη − 1 . W e p r o ve Theorem 1.1 with ǫ = 1 2 , δ = 2 − 500 , and c = 2 − 700 , and we mak e no attempt to optimize constan ts. The proof of Theorem 1.1 can b e easily modiﬁed to obtain th e same result w ith ǫ tending to 1 at the exp ense of having δ and c tending to 0. In the imp ortan t sp ecial case where ǫ = δ = η , w e let M ( ǫ ) = M ( ǫ, δ , η ). Go we rs [22 ] ga ve t wo diﬀeren t constructions giving lo wer b ounds on M ( ǫ ). The ﬁr st construction is simp ler, but the lo w er b ound it giv es is a to w er of tw os of heigh t only logarithmic in ǫ − 1 . The second construction gi v es a low er b ound wh ic h is a to wer of t wos of heigh t ǫ − 1 / 16 , b ut is more complicated. T heorem 1.1 also giv es a lo wer b ound on M ( ǫ ) whic h is a to w er of t w os of heigh t p olynomial in ǫ − 1 , in fact linea r in ǫ − 1 , and the construction is a bit simp ler. Un fortunately , the pro of that it works, whic h bu ilds u p on Go w ers’ simpler ﬁ r st pro of, is still rather complicated and d elicate . W e giv e a rough idea of how the graph G used to pr ov e Th eorem 1.1 is constructed. The graph G has a sequence of vertex partitions P 1 , . . . , P s , w ith P i +1 a reﬁnement of P i for 1 ≤ i ≤ s − 1, and the num b er of parts of P i +1 is r oughly exp onent ial in the n umb er of parts of P i . F or eac h i , 1 ≤ i ≤ s − 1, we pic k a r andom graph G i with v ertex set P i , where eac h edge is pick ed indep endent ly with probabilit y p i . F or ev ery t wo vertex sub sets X , Y ∈ P i of G whic h are adj acen t in G i , w e tak e random vertex partitions X = X 1 Y ∪ X 2 Y and Y = Y 1 X ∪ Y 2 X in to parts of equal size, with eac h of these parts the union of parts of P i +1 . Then, for d = 1 , 2, we add th e edges to G b et ween X d Y and Y d X . W e will sho w that with p ositiv e probabilit y the graph G co nstructed ab o v e has the desired prop erties for T h eorem 1.1. In fact, in Theorem 3.1, we will sho w that it has the stronger p rop erty that any ( ǫ, δ , η )-regular v ertex partition of G is close to b eing a reﬁnemen t of P s . A n o velt y of our construction, n ot pr esen t in the constructions of Go wers, is the use of the random graphs G i , which allo w us to control the n u mb er of irregular pairs. Instead, for ev ery pair of parts X, Y in P i , Go w ers [22] in tro du ces or deletes s ome edges b et ween th em so as to mak e the pair of parts far from regular. T o pro v e the desired r esu lt, w e ﬁr st establish sev eral lemmas on the edge distr ibution in G . The construction is general enough an d T heorem 3.1 strong enough that we also use it to establish a wo wzer-t yp e lo w er b ound for the strong regularit y lemma, as describ ed in the next sub section. 1.2 The str ong regularity lemma Before stating the strong regularity lemma, we n ext d eﬁ ne a n otion of closeness b et ween an equ itable partition and an equitable reﬁnement of this partition. F or an equitable partition A = { V i | 1 ≤ i ≤ k } of V ( G ) and an equitable reﬁnement B = { V i,j | 1 ≤ i ≤ k , 1 ≤ j ≤ ℓ } of A , we sa y that B is ǫ -close to A if the f ollo wing is satisﬁed. All 1 ≤ i ≤ i ′ ≤ k but at most ǫk 2 of them are su c h that for all 1 ≤ j, j ′ ≤ ℓ b u t at most ǫℓ 2 of th em | d ( V i , V i ′ ) − d ( V i,j , V i ′ ,j ′ ) | < ǫ holds. This n otion roughly sa y s 4 that B is an appr oximati on of A . W e are no w r eady to state the strong regularit y lemma of Alon, Fisc her, Krivel evic h , and Szege dy [5]. Lemma 1.1 (Strong regularit y lemma) F or every function f : N → (0 , 1) ther e exists a numb er S = S ( f ) with the fol lowing pr op erty. F or every gr aph G = ( V , E ) , ther e is an e quitable p artition A of the vertex set V and an e quitable r eﬁnement B of A with |B | ≤ S such that th e p artition A is f (1) -r e gular, the p artition B is f ( |A| ) -r e gular, and B is f (1) -close to A . The upp er b ound on S , the num b er of parts of B , that the pro of giv es is of w o wzer-t yp e, which is one lev el higher in the Ac kermann hierarc h y than the to wer fun ction. The tower f unction is deﬁned b y T (1) = 2, and T ( n ) = 2 T ( n − 1) for n ≥ 2. The wow zer function W ( n ) is deﬁned b y W (1) = 2 and W ( n ) = T ( W ( n − 1)). F or reasonable c h oices of the function f , which is the case for all known applications, such as those for whic h 1 /f is an in cr easing function whic h is at least a constant n umber of iterations of the loga rithm function, the upp er b oun d on S ( f ) is at least wo wzer in a p o wer of ǫ = f (1). Recall that M ( ǫ ), the n u m b er of parts required for Szemer ´ edi’s regularit y lemma, gro ws as a to wer of h eight a p ow er of ǫ − 1 . The p recise upp er b ound on the num b er of parts in the stron g regularit y lemma is deﬁned as follo ws . Let W 1 = M ( ǫ ) and W i +1 = M (2 f ( W i ) /W 2 i ). The pro of of the strong regularit y lemma [5] sho ws that S ( f ) = 51 2 ǫ − 4 W j with j = 64 ǫ − 4 satisﬁes the required prop erty . F or a partition P : V ( G ) = V 1 ∪ . . . ∪ V k of the vertex set of a graph G , the me an squar e density of P is d eﬁned by q ( P ) = X i,j d 2 ( V i , V j ) p i p j , where p i = | V i | / | V ( G ) | . This fun ction pla ys an imp ortan t role in the pr o of of Szemer ´ edi’s regularity lemma and its v arian ts. The strong r egularit y lemma giv es a regular partition A , and a reﬁn emen t B whic h is m u c h more regular and is close to A . F or equ itable partitions A and B with B a reﬁn emen t of A , the condition B is ǫ -close to A is equiv alen t, u p to a p olynomial change in ǫ , to q ( B ) ≤ q ( A ) + ǫ . Indeed, if B is ǫ -close to A , then q ( B ) ≤ q ( A ) + O ( ǫ ), while if q ( B ) ≤ q ( A ) + ǫ , then B is O ( ǫ 1 / 4 )-close to A . A v ersion of this statemen t is present in Lemma 3.7 of [5]. As it is suﬃicen t and more conv enien t to w ork with mean sq u are density instead of ǫ -closeness, we do so f r om no w on. Note that in the strong regularit y lemma, without loss of generalit y w e may assume f is a (mono- tonically) decreasing function. In deed, this can b e shown by considering the d ecreasing function f ′ ( k ) := min 1 ≤ i ≤ k f ( k ). F rom the ab ov e d iscussion, it is easy to see that the str ong regularit y lemma has the follo wing simple corollary , with a similar upp er b ound. Corollary 1.1 L et ǫ > 0 and f : N → (0 , 1) b e a de cr e asing function. Then ther e exists a numb er S = S ( f , ǫ ) such that for every gr aph G ther e ar e e quitable p artitions A , B of th e vertex set of G with |B | ≤ S , q ( B ) ≤ q ( A ) + ǫ , and B is f ( |A| ) -r e gular. W e prov e a lo we r b ound for the strong r egularit y lemma of wo wzer-t yp e, which essentiall y matc h es th e upp er b ound. Maybe surprisin gly , our construction fur ther sho ws th at m uch less than what is requir ed 5 from the s tr ong regularit y lemma already giv es wo wzer-t yp e b ounds. In particular, eve n f or Corollary 1.1, which app ears considerably wea k er than th e strong regularit y lemma, we get a wo wzer-t yp e lo wer b ound . Note that in Corollary 1.1, B is not requ ir ed to b e a r eﬁ nemen t of A . In this case we could ha ve q ( B ) b eing close to q ( A ) bu t the edge densities b et ween the parts in these partitions are quite diﬀeren t fr om eac h other, i.e., these partitions are not close to eac h other. Theorem 1.2 L et 0 < ǫ < 2 − 100 and f : N → (0 , 1) b e a de cr e asing func tion with f (1) ≤ 2 − 100 ǫ 6 . Deﬁne W ℓ r e cursively by W 1 = 1 , W ℓ +1 = T  2 − 70 ǫ 5 /f ( W ℓ )  , wher e T i s the tower function. L et W = W t − 1 with t = 2 − 20 ǫ − 1 . Then ther e is a gr aph G su c h that if e quitable p artitions A , B of the vertex set of G satisfy q ( B ) ≤ q ( A ) + ǫ and B is f ( |A| ) -r e gular, then |A| , |B | ≥ W . W e ha v e the follo wing corollary (by replacing ǫ by ǫ 1 / 7 ), which is a simple to state lo wer b ound of w o wzer-type. Corollary 1.2 F or 0 < ǫ < 2 − 700 , ther e is a gr aph G such tha t if e quitable p artitions A , B of the vertex set of G satisfy |B | ≥ |A| , q ( B ) ≤ q ( A ) + ǫ and B is ǫ/ | A | -r e gular, then |B | , |A | ar e b ounde d b elow by a function which is wowzer in Ω ( ǫ − 1 / 7 ) . 1.3 Induced graph remo v al Let H b e a ﬁxed graph on h v ertices and let G b e a graph with o ( n h ) copies of H . T o pro v e the graph remo v al lemma, we need to pro v e that all copies of H can b e remo ved from G by deleting o ( n 2 ) edges. The standard approac h is to apply the regularit y lemma to the graph G to obtain an ǫ -regular v ertex partition (with an ap p ropriate ǫ ) into a constan t num b er of parts M ( ǫ ). Then delete edges b et we en pairs of parts ( V i , V j ), including i = j , if the p air is n ot ǫ -regular or the density b etw een the pair is small. It is easy to see that there are few deleted edges. F urthermore, if there is a cop y of H in the remaining s ubgraph, then the edges go b et wee n pairs of p arts which are ǫ -regular and not of small densit y . A count ing lemma then s ho w s that in suc h a case th e num b er of copies of H is Ω( n h ) in the remaining subgraph, and hence in G as wel l. Bu t this w ould con tradict the assump tion that G has o ( n h ) copies of H , so all copies of H must already hav e b een remo ved. Recall that the indu ced graph remo v al lemma [5] is the analogous statement for indu ced subgraphs, and it is str onger than the graph remo v al lemma. It states that f or any graph H on h vertic es and ǫ > 0 there is δ = δ ( ǫ, H ) > 0 suc h that if a graph G on n v ertices has at most δ n h induced co pies of H , th en we can add or d elete ǫn 2 edges of G to obtain an induced H -fr ee graph. One well- kno wn applicatio n of the induced graph remov al lemma is in p rop ert y testing. This is an activ e area of computer science wh ere one wishes to quic kly distinguish b et ween ob jects th at satisfy a prop erty from ob j ects that are far from satisfying that prop ert y . The study of this notion w as initiated by R u binﬁeld and Sud an [35], and subsequently Goldreich, Goldw asser, and Ron [21] started the in v estigati on of prop ert y testers f or combinatoria l ob jects. One simple consequence of the in d uced graph remo v al lemma is a constan t time algorithm for ind uced subgraph testing with one-sided error (see [2] and its references). A graph on n v ertices is ǫ -far from b eing induced H -free if at least ǫn 2 6 edges need to b e added or remov ed to mak e it indu ced H -free. The induced graph remov al lemma implies th at th er e is an alg orithm whic h runs in time O ǫ (1) w hic h acce pts all in d uced H -fr ee graphs, and rejects an y graph wh ich is ǫ -far from b eing ind uced H -free with probabilit y at lea st 2/3. Th e algorithm samples t = 2 δ − 1 h -tuples of v ertices uniformly at random, where δ is pic k ed according to the in duced graph remo v al lemma, and accepts if n one of them form an induced cop y of H , and otherwise rejects. Any induced H -fr ee graph is clearly accepted. If a graph is ǫ -far from b eing ind u ced H -free, then it con tains at least δ n h copies of H , and the prob ab ility that none of the sampled h -tuples forms an in duced cop y of H is at most (1 − δ ) t < 1 / 3. Notice that the runn ing time as a function of ǫ dep ends on the b ound in th e indu ced graph remo v al lemma, an d the p r o of using th e strong r egularit y lemma giv es a wo w zer-t yp e dep en dence. It is temp tin g to try the same approac h u s ing S zemer´ edi’s regularit y lemma to obtain the induced graph remo v al lemma. Ho wev er, th er e is a sig niﬁcant problem with this appr oac h, which is handling the p airs b et we en irregular pairs. T o get around th is issue, Alon, Fisc her, Krivelevi c h , and Szegedy [5] d ev elop ed the strong regularity lemma. Because of its applications, in cluding those in graph p rop ert y testing, it has remained an int riguing problem to impro v e the b ound in the ind u ced graph remo v al lemma. Th is p roblem has b een discuss ed in sev eral pap ers b y Alon and his collaborators [2], [6], [8]. Th e main r esult d iscussed in this subsection addresses this p roblem, improving the b ou n d on the num b er of p arts in the ind uced graph remo v al lemma from w o wzer-t yp e to to w er-t yp e. The tower function t i ( x ) is deﬁned by t 0 ( x ) = x and t i +1 ( x ) = 2 t i ( x ) . W e sa y that t i ( x ) is a to wer in x of heigh t i . Theorem 1.3 F or any gr aph H on h v ertic es and 0 < ǫ < 1 / 2 ther e i s δ > 0 with δ − 1 a tower in h of height p olynomial in ǫ − 1 such that if a gr aph G on n ve rtic es has at most δn h induc e d c opies of H , then we c an add or delete ǫn 2 e dges of G to obta in an induc e d H -fr e e gr aph. The follo wing lemma is an easy corollary of the strong regularit y lemma which w as used in [5 ] to establish th e induced graph remo v al lemma. Lemma 1.2 F or e ach 0 < ǫ < 1 / 3 and de cr e asing function f : N → (0 , 1 / 3) ther e is δ ′ = δ ′ ( ǫ, f ) such that every gr aph G = ( V , E ) with | V | ≥ δ ′− 1 has an e quitable p artition V = V 1 ∪ . . . ∪ V k and v ertex subsets W i ⊂ V i such tha t | W i | ≥ δ ′ | V | , e ach p air ( W i , W j ) with 1 ≤ i ≤ j ≤ k i s f ( k ) -r e gular, and al l but at most ǫk 2 p airs 1 ≤ i ≤ j ≤ k satisfy | d ( V i , V j ) − d ( W i , W j ) | ≤ ǫ . In fact, Lemma 1.2 is a little b it stronger than the original v ersion in [5] in that eac h set W i is f ( k )-regular with itself. Th e orig inal version follo ws fr om the strong regularit y lemma b y taking the partition V = V 1 ∪ . . . ∪ V k to b e the partiton A in the s tr ong r egularity lemma, and the sub s et W i to b e a random part V ij ⊂ V i of the reﬁnemen t B of A in th e strong regularit y lemma. F r om this sligh tly s tr onger v ersion, the pro of of the indu ced graph r emo v al lemma is a bit simpler and shorter. Ind eed, w ith f ( k ) = 1 4 h ǫ h , which do es n ot dep end on k , if there is a m apping φ : V ( H ) → { 1 , . . . , k } suc h that for all adjacen t v ertices v , w of H , the edge densit y b et wee n W φ ( v ) and W φ ( w ) is at least ǫ , and for all distinct nonadjacen t ve rtices v , w of H , the edge den s it y b et wee n W φ ( v ) and 7 W φ ( w ) is at most 1 − ǫ , then a standard counting lemma shows that G con tains at least δn h induced copies of H , where δ = ( ǫ/ 4) ( h 2 ) δ ′ h . Hence, we may assume that there is no suc h mapping φ . W e then delete ed ges b et we en V i and V j if the edge densit y b et wee n W i and W j is less than ǫ , and one adds the edges betw een V i and V j if the d ensit y b et wee n W i and W j is more than 1 − ǫ . The total num b er of edges added or remov ed is at m ost 5 ǫn 2 , and no indu ced cop y of H remains. Replacing ǫ by ǫ/ 8 in the ab o ve argumen t give s the indu ced graph remov al lemma. W e ﬁ n d another pro of of Lemma 1.2 with a b etter to wer-t yp e b ound. This in turn implies, by the argument sk etc hed ab o v e, the to w er-t yp e b ound for the in duced graph remo v al lemma stated in Theorem 1.3 . The starting p oin t for our app roac h to L emma 1.2 is a w eak r egularit y lemma d ue to Duk e, Lefmann and R¨ odl [15]. This lemma sa y s that for a k -partite graph, b etw een sets V 1 , V 2 , . . . , V k , there is an ǫ -regular p artition of th e cylinder V 1 × · · · × V k in to a r elativ ely small num b er of cylinders K = W 1 × · · · × W k , w ith W i ⊂ V i for 1 ≤ i ≤ k . The deﬁ n ition of an ǫ -r e gular partition here is that all but an ǫ -fraction of the k -tup les ( v 1 , . . . , v k ) ∈ V 1 × · · · × V k are in ǫ -r e gular cylinders, where a cylinder W 1 × · · · × W k is ǫ -regular if all  k 2  pairs ( W i , W j ), 1 ≤ i < j ≤ k , are ǫ -regula r in the u sual sense. In the same w a y that one d eriv es the strong regularit y lemma from the ordinary regularit y lemma, w e show ho w to d eriv e a strong version of this lemma. W e will r efer to this strengthenin g, of whic h Lemma 1.2 is a straigh tforward consequence, as the str ong cylinder r e gularity lemma . It will also b e con venien t if, in th is lemma, we mak e the requirement that a cylinder b e r egular slightly stronger, b y asking that ea c h W i b e regular with itself. T hat is, we sa y that a cylinder W 1 × · · · × W k is str ongly ǫ -r e gular if all pairs ( W i , W j ) with 1 ≤ i, j ≤ k are ǫ -r egular. A partition K of the cylind er V 1 × · · · × V k in to cylind ers K = W 1 × · · · × W k , with W i ⊂ V i for 1 ≤ i ≤ k , is then said to str ongly ǫ -r e gular if all but an ǫ -fractio n of the k -tuples ( v 1 , . . . , v k ) ∈ V 1 × · · · × V k are in strongly ǫ -regular cyli nders. Let P : V = V 1 ∪ · · · ∪ V k b e a partitio n of the v ertex set of a graph and K b e a partition of the cylinder V 1 × · · · × V k in to cylinders. F or eac h K = W 1 × · · · × W k , w ith W i ⊂ V i for 1 ≤ i ≤ k , w e let V i ( K ) = W i . W e then deﬁne the p artition Q ( K ) o f V to b e the reﬁnemen t of P wh ic h is the common reﬁnement of all the parts V i ( K ) w ith i ∈ [ k ] and K ∈ K . The strong cylinder r egularity lemma is no w as follo w s. Lemma 1.3 F or 0 < ǫ < 1 / 3 , p ositive inte ger s , and de cr e asing f u nction f : N → (0 , ǫ ] , ther e i s S = S ( ǫ, s, f ) such that the fol lowing holds. F or every gr aph G , ther e is an inte ger s ≤ k ≤ S , an e qui table p artition P : V = V 1 ∪ . . . ∪ V k and a str ongly f ( k ) -r e gular p artition K of the c ylinder V 1 × · · · × V k into cylinders satisfying that the p artition Q = Q ( K ) of V has at most S p arts and q ( Q ) ≤ q ( P ) + ǫ . F urthermor e, ther e is an absolute c onstant c su c h that letting s 1 = s and s i +1 = t 4 (( s i /f ( s i )) c ) , we may take S = s ℓ with ℓ = 2 ǫ − 1 + 1 . In ord er to pro ve th is lemma, we need, in addition to the Duke -Lefmann-R¨ odl regularit y lemma, a lemma sh owing that f or eac h ǫ > 0 there is δ > 0 su c h that ev ery graph G = ( V , E ) con tains a ve rtex subset U with | U | ≥ δ | V | whic h is ǫ -regular with itself, where, crucially , δ − 1 is b ounded ab o ve by a to wer function of ǫ − 1 of ab s olute constan t heigh t. While seemingly standard, we d o not kno w of suc h a resu lt in the literature. 8 Lemma 1.2 follo ws from Lemma 1.3 b y considering a random cylinder K in the cylinder partition K , with eac h cylinder pic ke d with pr obabilit y p rop ortional to its size, and letting W i = V i ( K ). 1.4 F rieze-Kannan w eak regularit y lemma F r ieze and Kannan [19], [20] dev elop ed a we ak er notion of regularit y wh ic h is suﬃcien t for certain applications and for wh ic h th e dep endence on the ap p ro x im ation ǫ is m uc h b etter. It states the existence of a verte x partitio n into a small n umb er of p arts for wh ic h the num b er of edges across an y t wo verte x subsets is within ǫn 2 of what is expected based on the ed ge densities b et ween the parts of the p artition and the intersecti on sizes of the v ertex su bsets with these parts. Lemma 1.4 (F rieze-Kannan w eak regularit y lemma) F or e ach ǫ > 0 ther e is a p ositive inte ger k ( ǫ ) such that every gr aph G = ( V , E ) has an e qu i table vertex p artition V = V 1 ∪ . . . ∪ V k with k ≤ k ( ǫ ) satisfying that for al l sub se ts A, B ∈ V , we ha ve   e ( A, B ) − X 1 ≤ i,j ≤ k d ( V i , V j ) | A ∩ V i || B ∩ V j |   ≤ ǫ | V | 2 . The w eak r egularit y lemma has a num b er of algorithmic applications. F r ieze and Kann an [20] u sed the w eak regularit y lemma to giv e constant- time appro ximation algo rithms for some general problems in dense graph s, a sp ecial case b eing the Max-Cut of a graph. Rece nt ly , Bansal and Williams [12] used the w eak regularit y lemma to obtain a faster com binatorial algorithm for Bo olean matrix m ultiplication. The imp ortance of the w eak regularity lemma is further discussed in th e citation of the r ecen t Knuth Prize to Kannan. As th er e are sev eral applications of the w eak regularit y lemma to fundamen tal algorithmic problems, w e would lik e to kno w the correct b ounds on the num b er of parts f or the wea k regularit y lemma. The pro of of the weak r egularit y lemma [20] sho ws that we ma y tak e k ( ǫ ) = 2 O ( ǫ − 2 ) . If this upp er boun d could b e impro ved, it w ould lead to faster algorithms for several problems of inte rest. Lo v´ asz and Szegedy [29] s tudied the problem of estimating the minimum num b er of parts k ( ǫ ) required for the w eak regularit y lemma, pro vin g a lo w er b oun d on k ( ǫ ) of the form 2 Ω( ǫ − 1 ) . Here we close the gap by pro ving a new lo wer b ound whic h matc hes the up p er b ound . Theorem 1.4 F or e ach ǫ > 0 , ther e ar e gr aphs for which the minimum numb er of p arts in a we ak r e gular p artition with appr oximation ǫ is 2 Ω( ǫ − 2 ) . A careful analysis of the pro of of T heorem 1.4 sho w s that the num b er of parts required in the wea k regularit y lemma with appr o ximation ǫ is at lea st 2 − 2 − 60 ǫ − 2 for 0 < ǫ ≤ 2 − 50 . In fact, the th eorem yields a stronger result, since we do not h ere require that the partition b e equitable. While the n u m b er of parts in the weak regularity lemma is 2 Θ( ǫ − 2 ) , the pro of obtains the partition as an o v er lay of only O ( ǫ − 2 ) sets. As discussed in [29], in some applications, such as in [4], this can b e treated as if there were only ab ou t O ( ǫ − 2 ) classes, which mak es the wea k regularity lemma quite eﬃcien t. It was sho w n in [4], and is also implied by Theorem 1.4, that the partition cannot b e th e o verla y of fewer sets. 9 1.5 The regular appro ximation lemma Another strengthenin g of S zemer ´ edi’s regularit y lemma came from the study of graph limits by Lo v´ asz and S zegedy [29], and also from w ork on the hyp ergraph generalizatio n of the r egularit y lemma b y R¨ odl and Sc hac ht [32]. Th is regularit y lemma, kno wn as the r egular appr o ximation lemma [3 3], pr o vides an arbitrary p recision for the regularit y as a fun ction of the n umb er of parts of the partition if an ǫ -fraction of the edges are allo wed to b e added or remo v ed. F or a function g : N → (0 , 1), a partition of the v ertex set into k parts is g -r e gular if all p airs of d istinct parts in the partition are g ( k )-regular. Lemma 1.5 ( Regular appro ximation lemma ) F or every ǫ > 0 , p ositive inte ger s and de cr e asing function g : N → (0 , 1) , ther e is an inte ger T = T ( g , ǫ, s ) so that given a gr aph G with n vertic es, one c an ad d-to/r emove-fr om G at most ǫn 2 e dges and thus get a gr aph G ′ that has a g -r e gular e quitable p artition of or der k for some s ≤ k ≤ T . Lo v´ asz and Szegedy [29] state that the regular app ro ximation lemma is equiv alen t to th e strong regularit y lemma, Lemma 1.1. It is not diﬃcult to dedu ce Lemma 1.5 from th e strong regularit y lemma, see [9] or [33] for d etails. Unlik e the original graph limit appr oac h, this p ro of of the regular appro ximation lemma gi v es explicit b ounds and yields a p olynomial time alg orithm for ﬁnding the partition and the necessary edge mo diﬁcations. In the other direction, b y applying Lemma 1.5 with 1 /g a to wer in the 1 /f f rom L emma 1.1, letting A b e the g -r egular partition of G ′ , and then using Szemer ´ edi’s regularit y lemma to get a r eﬁ nemen t B o f A wh ic h is an f ( A )-regular partition of G , it is easy to deduce the stron g regularit y lemma. The ma jor ca vea t here is the additional use of S zemer ´ edi’s regularit y lemma in d educing the strong regularit y lemma f rom the r egular appr o ximation lemma. Due to the add itional use of Szemer ´ edi’s regularit y lemma, it do es not r u le out the p ossibilit y that th e wo w zer-t yp e upp er b ound on T in the regular app ro xim ation lemma can b e impro v ed to to wer-t yp e. Ma yb e surprisingly , we indeed mak e suc h an impro vemen t. Theorem 1.5 F or ǫ > 0 , p ositive i nte ge r s and a de cr e asing f unction g : N → (0 , 1) , let δ ( t ) = min( g ( t ) 3 32 t 2 , ǫ/ 2) . L et t 1 = s and for i ≥ 1 let t i +1 = t i k ( δ ( t i )) , wher e k is as in the we ak r e gularity lemma, so k ( α ) = 2 O ( α − 2 ) . L et T 0 = t j with j = 4 ǫ − 2 . Then the r e gular appr oximation lemma holds with T = 16 T 0 /δ ( T 0 ) 2 . In other wor ds, the r e gular appr oximation lemma holds with a tower-typ e b ound. It is usually the case that 1 /g ( t ) in the regular approximat ion lemma is at most a to we r of constant heigh t in ǫ − 1 and t , and in this case the u pp er b oun d T on the num b er of parts is only a to we r of heigh t p olynomial in ǫ − 1 . Only in the unusual case of 1 /g b eing of tow er-t yp e gro w th do es th e num b er of p arts needed in the regular app ro ximation lemma gro w as wo wzer-t yp e. Alon, Sh apira, and Stav [9] giv e a pr o of of the regular approximat ion lemma w hic h yields a p olynomial time algorithm for ﬁ nding the partition and the necessary edge mo d iﬁcations. Similarly , our new pro of 10 can b e made algorithmic with a p olynomial time algorithm for ﬁnding the partition and the n ecessary edge modiﬁ cations. Making the pro of algo rithmic is essen tially the same as d on e in [9 ], so w e do not include the details. A partition of a graph sati sfying the weak r egularity lemma, Lemma 1.4 , is cal led a wea k ǫ -regular partition. T ao sh o wed [41] (see also [33]), by iterating th e w eak regularit y lemma, that one obtains the follo wing regularit y lemma wh ic h easily implies Szemer ´ edi’s regularit y lemma with the usu al to w er- t yp e b ounds. Lemma 1.6 F or al l ǫ > 0 , p ositive inte gers s and functions δ : N → (0 , 1) , ther e is a T 0 such that every gr aph has an e quitable ve rtex p artition P into t ≥ s p arts which is we ak ǫ -r e gular, an e q uitable vertex r e ﬁnement Q into at mo st T 0 p arts which is we ak δ ( t ) - r e gu lar, and q ( Q ) ≤ q ( P ) + ǫ . Let t 1 = s , and for i ≥ 1, let t i +1 = t i k ( δ ( t i )), where k is as in the weak regularit y lemma. Recal l k ( ǫ ) is exp onential in ǫ − 2 . T hen T 0 in Lemma 1.6 is giv en b y T 0 = t j with j = ǫ − 1 . In particular, if δ − 1 is b ound ed ab ov e b y a to wer of constant h eigh t, then T 0 in T ao’s regularit y lemma gro ws as a to wer of heigh t linear in ǫ − 1 . Our pr o of of Theorem 1.5 sho ws that the regular appr o ximation lemma is equiv alen t to T ao’s regularit y lemma with similar b ounds. In fact, we show that T in the regular appro ximation lemma can b e tak en to b e T = 16 T 0 /δ ( T 0 ) 2 , where T 0 = T 0 ( δ , ǫ 0 , s ) is the b ound on the n um b er of parts in T ao’s regularit y lemma, δ ( t ) = min( g ( t ) 3 32 t 2 , ǫ/ 2), and ǫ 0 = ( ǫ/ 2) 2 . As T ao’s regularit y lemma is a simple consequence of the regular appro ximation lemma and an application of the w eak regularit y lemma, it suﬃces to show ho w to ded uce the regular appro ximation lemma from T ao’s regularit y lemma. The pro of starts by applyin g T ao’s regularit y lemma with δ and ǫ 0 as ab o ve . F or eac h pair ( X , Y ) of parts in Q , where X ⊂ A and Y ⊂ B with A, B p arts of P , we randomly add/delete edges b et w een X, Y with a certain probability so that the densit y b et ween X and Y is ab out the same as the d ensit y b et w een A and B . W e sho w that in doing this w e hav e mad e ev ery pair ( A, B ) of p arts of P g ( t )- regular w ith t = | P | . Since q ( Q ) ≤ q ( P ) + ǫ 0 , the edge d ensit y d ( X, Y ) b et w een m ost pairs ( X, Y ) of parts of Q is close to the edge den sit y d ( A, B ) b et ween A and B , and few edges are changed to obtain a graph G ′ for which the p artition P is g -r egular. W e next br ieﬂy discuss low er boun ds for the regular appro ximation lemma. In the ca se g is a (small) constan t function, a to wer-t yp e lo we r b ound follo ws fr om T heorem 1.1. If g is at least a tow er fu nction, w e get a lo wer b ound of wo wzer-t yp e f rom Theorem 1.2 and the fact that the s trong regularit y lemma follo ws fr om the regularit y appro ximation lemma with an additional application of Szemer´ edi’s regularit y lemma as discussed earlier. On e could lik ely come up with a co nstruction giving a general lo wer b ound essen tially matc hing Theorem 1.5, but as the already men tioned int eresting cases discussed ab o v e are h andled by Th eorems 1.1 a nd 1.2, we do not include suc h a construction. Organization In the next section, we prov e some usefu l to ols for establishin g lo wer b ounds for Szemer ´ edi’s r egularit y lemma and th e strong r egularit y lemma. In Section 3, w e gi v e a general constru ction and us e it to pro v e Theorem 1.1 whic h addresses questions of Szemer ´ edi and Go wers on the n um b er of irr egular 11 pairs in Szemer´ edi’s regularit y lemma. In Section 4, w e u se the general construction to pro v e T h eorem 1.2, which gives a w o wzer-type lo w er b ound on the n umber of parts of the tw o partitions in the strong regularit y lemma. In Sectio n 5, w e pro v e the strong cylinder regularit y lemma and us e it to pro ve a to wer-t yp e upp er b ound on the ind uced graph r emo v al lemma. In Section 6, we pro ve a to wer-t yp e upp er b ound on the num b er of parts in the regular appro ximation lemma. In Section 7, we pro ve a tigh t lo wer b ound on the n um b er of p arts in the w eak regularit y lemma. These later sections, Sections 5, 6 and 7, are largely indep enden t of earlier sections and of eac h other. Th e in terested r eader ma y therefore skip forw ard without fear of losing the thread. W e ﬁnish with some co ncluding remarks. Th is includes a discussion sho wing that in the regularit y lemma, the condition that the p arts in the partition are of equal size d o es not aﬀect the b ounds by m uc h . W e also discuss an early v ersion of Szemer ´ edi’s regularit y lemma, and a recent result of Mallia ris and Shelah wh ic h shows an in teresting connection b et w een irregular pairs in the regularit y lemma and the app earance of h alf-graphs. Throughout the pap er, we systematically omit ﬂ o or and ceiling signs whenever they are not crucial for the sake of clarit y of presenta tion. W e also do not mak e any s erious attempt to optimize absolute constan ts in our statemen ts and pro ofs. 2 T o ols Supp ose S = S 1 + · · · + S n is the su m of n m utually indep endent random v ariables, where for eac h i , Pr[ S i = 1] = p and Pr[ S i = 0] = 1 − p . Th e s u m S h as a binomial d istribution w ith parameters p and n , and h as exp ected v alue pn . A Chernoﬀ-t yp e estimate (see Theorem A.1.4 in [10]) implies that for a > 0, Pr[ S − pn > a ] < e − 2 a 2 /n (1) By sym m etry , w e also ha v e Pr [ S − pn < − a ] < e − 2 a 2 /n and hen ce Pr[ | S − pn | > a ] < 2 e − 2 a 2 /n . W e start b y p ro vin g a couple of lemmas on the edge distribution of random bipartite graphs with diﬀeren t part sizes. C onsider the r an d om bipartite graph B = B ( m, M ) with parts [ m ] and [ M ] formed by eac h v ertex i ∈ [ m ] ha ving exactly M / 2 neighbors (we assu me M is even) in [ M ] p ic ked uniformly at random and indep endent ly o f the c hoices of the neigh b orho od s for the other v ertices in [ m ]. The follo wing lemma sho w s that, with high p robabilit y , certain simple estimates on the n u m b er of common n eigh b ors or non n eigh b ors of any t w o vertic es in B ( m, M ) hold. Lemma 2.1 L et M ≥ m b e p ositive inte gers with M ≥ 2 20 even, and 0 < µ < 1 / 2 b e such that m ≥ 2 µ − 2 log M . Then, with pr ob ability at le ast 1 − M − 2 , the r andom bip artite gr aph B = B ( m, M ) has the fol lowing pr op erties: • for any distinct j, j ′ ∈ [ M ] , the nu mb er of i for which j and j ′ ar e either b oth neighb ors of i or b oth nonneighb ors of i is less than ( 1 2 + µ ) m . 12 • for any distinct i, i ′ ∈ [ m ] , the numb er of c ommon neighb ors of i and i ′ and the numb er of c ommon nonneighb ors of i and i ′ in [ M ] ar e b oth less than  1 4 + M − 1 / 4  M . Pro of: Fix d istinct j, j ′ ∈ [ M ]. F or eac h i ∈ [ m ], the probabilit y th at i is adjacen t to b oth j, j ′ or nonadjacen t to b oth j, j ′ is ( M 2 − 1) / ( M − 1) < 1 2 , and these eve n ts are indep endent of eac h other. Therefore, b y (1), the prob ab ility that the num b er of i for whic h j and j ′ are either b oth neigh b ors of i or b oth nonneigh b ors of i is at least ( 1 2 + µ ) m is at most e − 2( µm ) 2 /m = e − 2 µ 2 m ≤ M − 4 . As th er e are  M 2  c hoices for j, j ′ , and 1 2 M − 2 ≥ M − 4  M 2  , by the union b ound we ha v e that B has the ﬁrst d esired prop erty w ith probabilit y at least 1 − 1 2 M − 2 . As the hypergeometric d istribution is at least as concent rated as the corresp on d ing binomial d istri- bution (for a pro of, s ee Section 6 of [25]), w e can app ly (1) to conclud e that for eac h ﬁxed pair i, i ′ ∈ [ m ] of distinct ve rtices the p r obabilit y that the num b er of common neighbors of i and i ′ is at least  1 4 + M − 1 / 4  M is at most e − 2( M − 1 / 4 M ) 2 / M = e − 2 M 1 / 2 . Similarly , for eac h ﬁxed pair i, i ′ ∈ [ m ] of distinct ve rtices the probability that the n umb er of common nonneigh b ors of i and i ′ in [ M ] is at least  1 4 + M − 1 / 4  M is at most e − 2 M 1 / 2 . As there are  m 2  c hoices for i, i ′ and 1 2 M − 2 ≥ 2 e − 2 M 1 / 2  m 2  , by the union b ound we ha ve that B has the second desired p r op ert y with probabilit y at least 1 − 1 2 M − 2 . Hence, with pr obabilit y at least 1 − M − 2 , B h as b oth d esired pr op erties, whic h completes the pr o of. ✷ The next lemma sho ws that the edges in B ( m, M ) are almost surely uniform ly distribu ted b et ween large vertex subsets. Lemma 2.2 L et M and m b e p ositive inte gers with M even. With pr ob ability at le ast 1 − M − 1 , for any U 1 ⊂ [ m ] and U 2 ⊂ [ M ] with | U 1 | = u 1 and | U 2 | = u 2 , we have | e B ( U 1 , U 2 ) − 1 2 u 1 u 2 | ≤ p f , (2) wher e f = f ( u 1 , u 2 ) = u 1 u 2  u 1 ln em u 1 + u 2 ln eM u 2  . Pro of: F or ﬁxed su b sets U 1 ⊂ [ m ] and U 2 ⊂ [ M ], the random v ariable e B ( U 1 , U 2 ), wh ic h h as mean 1 2 | U 1 || U 2 | , desp ite not satisfying a bin omial distribu tion, still satisﬁes the estimate (1) for th e corresp onding b in omial distribution with parameters 1 / 2 and | U 1 || U 2 | . Indeed, note that e B ( U 1 , U 2 ) is th e sum of the degrees of the v ertices of U 1 in U 2 , and these | U 1 | d egrees are identic al indep endent random v ariables, eac h sati sfying a h yp ergeometric d istribution. By T heorem 4 in S ection 6 of [25], the exp ected v alue of the exp onentia l of a random v ariable with a hyp er geometric distrib ution is at most the exp ected v alue of the exp onen tial of the rand om v ariable with the corresp ondin g b in omial distribution. Su bstituting this esti mate in to the pro of of (1) shows that the Cher n oﬀ estimate also holds for e B ( U 1 , U 2 ). Hence, the probability (2 ) do esn’t hold for a particular pair U 1 , U 2 is less than 13 2 e − 2 f / ( u 1 u 2 ) . By the u nion b oun d , the probabilit y th at there is a p air of su bsets U 1 ⊂ [ m ] and U 2 ⊂ [ M ] not satisfying (2) is at most m X u 1 =1 M X u 2 =1  m u 1  M u 2  2 e − 2 f / ( u 1 u 2 ) ≤ m X u 1 =1 M X u 2 =1  em u 1  u 1  eM u 2  u 2 2 e − 2 f / ( u 1 u 2 ) = m X u 1 =1 M X u 2 =1 2  em u 1  − u 1  eM u 2  − u 2 ≤ M − 1 . ✷ F r om the bipartite graph B , w e construct equitable p artitions ( A i , B i ) m i =1 of [ M ], by letting A i denote the set of neigh b ors of vertex i ∈ [ m ] in graph B . F r om Lemmas 2.1 and 2.2, we ha ve the follo wing corollary . Corollary 2.1 Supp ose M ≥ m ar e p ositive inte gers with M ≥ 2 20 even, and 0 < µ < 1 / 2 is such that m ≥ 2 µ − 2 log M . Ther e is a bip artite gr aph B with p arts [ m ] and [ M ] , with e ach vertex in [ m ] of de g r e e M / 2 with the fol lowing pr op erties. The estimate (2) holds for al l U 1 ⊂ [ m ] and U 2 ⊂ [ M ] with | U 1 | = u 1 and | U 2 | = u 2 , and B satisﬁes the two pr op erties in the c onclusion of L emma 2.1 . The next lemma is a u seful co nsequence of the equitable partitions ( A i , B i ) m i =1 b eha ving randomly . Giv en a ve ctor λ ∈ R M and 1 ≤ q < ∞ , write || λ || q for  P M i =1 | λ i | q  1 /q and || λ || ∞ for max 1 ≤ i ≤ M | λ i | . Lemma 2.3 L et M b e a p ositive even inte g e r, 0 < µ < 1 / 2 , and ( A i , B i ) m i =1 b e a se q uenc e of p artitions satisfying the c onclusion of Cor ol lary 2.1. Supp ose that 0 < σ , τ , α ar e such that σ , τ < 1 , α < 1 / 2 , and ( 1 2 − µ )(1 − σ 2 ) > τ 2 + 2(1 − τ ) α (1 − α ) . Then for every se q u enc e λ = ( λ 1 , . . . , λ M ) of nonne gative r e al numb ers which ar e not al l zer o with || λ || 2 = σ || λ || 1 , ther e ar e at le ast τ m values of i for which min( a i , b i ) > α || λ || 1 , wher e a i = P j ∈ A i λ j and b i = P j ∈ B i λ j . Pro of: Note that by m ultiplying ea c h λ j b y 1 / || λ || 1 , we ma y assume without lo ss of generalit y that || λ || 1 = 1. F or distinct j, j ′ ∈ [ M ], let ( j, j ′ ) i denote that j and j ′ lie in diﬀerent sets in the partition ( A i , B i ). Since for any distinct j, j ′ ∈ [ M ], th e num b er of i for which ( j, j ′ ) i holds is at least ( 1 2 − µ ) m , w e hav e X ( j,j ′ ) i λ j λ j ′ ≥ ( 1 2 − µ ) m X j λ j (1 − λ j ) = ( 1 2 − µ ) m ( || λ || 1 − || λ || 2 2 ) = ( 1 2 − µ ) m (1 − σ 2 ) , (3) where the sum is o ver all or der e d triples ( j, j ′ , i ) with j, j ′ distinct and j and j ′ lie in diﬀerent s ets in the p artition ( A i , B i ). W e ha ve the identit y X ( j,j ′ ) i λ j λ j ′ = 2 X i a i b i . 14 Since a i + b i = 1, w e hav e a i b i ≤ 1 / 4 and if m in( a i , b i ) ≤ α , then a i b i ≤ α (1 − α ). So if min( a i , b i ) ≤ α for all but less than τ m v alues of i , then X ( j,j ′ ) i λ j λ j ′ < τ 2 m + 2(1 − τ ) mα (1 − α ) . Comparing with (3) and dividing b y m , th is con tradicts the sup p osition, and completes the pro of. ✷ As usu al, G ( n, p ) denotes the r an d om graph on n vertic es c hosen by pic king eac h pair of ve rtices as an edge rand omly and indep enden tly with probabilit y p . W e ﬁn ish this section with a few standard lemmas on the edge d istr ibution in G ( n, p ). Lemma 2.4 In G ( n, p ) , with pr ob ability at le ast 1 − n − 2 , e very p air of disjoint vertex sub se ts U 1 and U 2 satisfy | e ( U 1 , U 2 ) − pu 1 u 2 | ≤ √ g , (4) wher e u 1 = | U 1 | , u 2 = | U 2 | and , for u 1 ≤ u 2 , g = g ( u 1 , u 2 ) = 2 u 1 u 2 2 ln ne u 2 . Pro of: F or ﬁxed sets U 1 and U 2 , the qu an tit y e ( U 1 , U 2 ) is a b inomial distributed rand om v ariable with parameters u 1 u 2 and p . By (1), we ha ve th at the probabilit y (4) do es not h old is less th an 2 e − 2 g / ( u 1 u 2 ) . By the un ion b ound, th e pr obabilit y that there are d isjoin t sets U 1 and U 2 for wh ic h (4) do es not hold is at most n X u 2 =1 u 2 X u 1 =1  n u 2  n − u 2 u 1  2 e − 2 g / ( u 1 u 2 ) ≤ n X u 2 =1 u 2 X u 1 =1  ne u 2  u 2  ne u 1  u 1 2 e − 2 g / ( u 1 u 2 ) ≤ n X u 2 =1 u 2 X u 1 =1 2  ne u 2  − 2 u 2 ≤ n − 2 . The resu lt follo ws. ✷ Lemma 2.5 In G ( n, p ) , with pr ob ability at le ast 1 − n − 2 , every vertex subset U satisﬁes | e ( U ) − p  u 2  | ≤ √ g , (5) wher e u = | U | and g = g ( u ) = 1 2 u 3 ln ne u . Pro of: F or ﬁxed U , the quan tit y e ( U ) is a binomially distribu ted r andom v ariable with parameters  u 2  and p . By (1), w e h a ve that the probabilit y (5) do es n ot hold is less than 2 e − 2 g / ( u 2 ) . By the union b ound , th e probabilit y that there is a v ertex subset U for which (5) do es n ot hold is at most n X u =2  n u  2 e − 2 g / ( u 2 ) ≤ n X u =2  ne u  u 2 e − 2 g / ( u 2 ) ≤ 2 n X u =2  ne u  − ( u +1) ≤ n − 2 . ✷ Com b ining the estimates from the pr evious t wo lemmas, we can b ound the pr obabilit y in G ( n, p ) th at there are t wo not necessarily disjoin t subsets with large ed ge discrepancy b etw een th em. 15 Lemma 2.6 In G ( n, p ) , the pr ob ability that ther e ar e inte gers u 1 and u 2 with u 1 ≤ u 2 and not ne c es- sarily disjoint vertex subsets U 1 and U 2 with | U 1 | = u 1 and | U 2 | = u 2 such that | e ( U 1 , U 2 ) − pu 1 u 2 | > 5 √ h, (6) wher e h = h ( u 1 , u 2 ) = u 1 u 2 2 ln ne u 2 , is at mos t 2 n − 2 . Pro of: F or sets U 1 and U 2 , letting U ′ 1 = U 1 \ U 2 , U ′ 2 = U 2 \ U 1 , and U = U 1 ∩ U 2 , we h a ve e ( U 1 , U 2 ) = e ( U ′ 1 , U 2 ) + 2 e ( U ) + e ( U, U ′ 2 ) . W e ha v e that the b ounds in Lemmas 2.4 and 2.5 hold with prob ab ility at least 1 − 2 n − 2 . Hence, usin g the triangle inequalit y , and | U ′ 1 | ≤ u 1 , | U | ≤ u 1 , | U ′ 2 | ≤ u 2 , we ha v e | e ( U 1 , U 2 ) − pu 1 u 2 | ≤ 2 p g ( u 1 , u 2 ) + 2 p g ( u 1 ) + pu 1 ≤ 5 √ h with probabilit y at least 1 − 2 n − 2 . Here the extra pu 1 factor comes fr om the fact that degenerat e edges are not count ed in e ( U ). ✷ 3 A general graph construction In this section, w e will d eﬁ ne a nonuniform random graph G = ( V , E ) which, assuming certain es- timates, has th e pr op ert y that an y suﬃcien tly regular partition of its v ertex set is close to b eing a reﬁnement of a particular partition of G in to man y parts. As this particular partition has many parts, this will imp ly th at any suﬃ cien tly regular partition will ha v e man y parts. After deﬁning G , we w ill pro v e that certain usefu l estimates on the edge d istribution of G hold with p ositiv e probab ility . W e will u se these estimates to show that G has the desired pr op ert y . 3.1 Deﬁning gr aph G F ollo wing Go we rs [22], w e attempt to rev erse engineer th e pr o of of Szemer ´ edi’s regularit y lemma to sho w that the upp er b ound is essen tially b est p ossible. The pro of of the regularit y lemma follo ws a sequence of reﬁn emen ts of th e v ertex set of th e graph u n til w e arrive at a r egular partitio n, with the num b er of p arts in eac h partition exp onen tially larger than in the pr evious partition. W e build a sequence of partitions of th e v ertex set, and then describ e h o w the edges of G are distributed b et w een the v arious parts of th e partition. T o show that any (suﬃciently) regular partition Z of V ( G ) requires man y p arts, we sh o w that Z is roughly a reﬁnemen t of the p artitions we constructed in deﬁning G . Let m 1 ≥ 2 200 b e a p ositiv e in teger and ρ = 2 − 20 . F or 2 ≤ i ≤ s , let m i = m i − 1 a i − 1 , where a i − 1 = 2 ⌊ ρm 9 / 10 i − 1 ⌋ . S upp ose p i ≥ m − 1 / 10 i for 1 ≤ i ≤ s − 1. The verte x set V has a sequence of equitable p artitions P 1 , . . . , P s , wh ere P j is a reﬁnemen t of P i for j > i deﬁned as follo w s. The num b er of parts of P i is m i . F or ea c h set X in partition P i , we pic k 16 an equitable partitio n of X in to a i parts, and let P i +1 b e the partition of V with m i +1 = m i a i parts consisting of the union of th ese partitions of parts of P i . F or 1 ≤ i ≤ s − 1, let G i b e a un iform random graph on P i with edge probability p i . Th at is, the v ertices of th e graph are the m i pieces of the partition and we p lace edges indep end en tly with probabilit y p i . In p ractice, w e w ill mak e certain sp eciﬁc assumptions ab out the edge d istribution of G i but these will hold with high p robabilit y in a random graph. F or example, w e shall assu me that ev ery v ertex in G i has degree at least p i m i / 2. F or eac h X , Y ∈ P i with ( X, Y ) an edge of G i , w e ha ve an equita ble partition Q X Y : X = X 1 Y ∪ X 2 Y in to t wo parts, where X j Y is a union of some of the parts in P i +1 for j = 1 , 2. F or eac h X ∈ P i , we shall c ho ose the partitions Q X Y with Y adjacen t to X in G i to satisfy the prop erties of Corollary 2.1 w ith µ = 2 ρ 1 / 2 = 2 − 9 . Note that this is p ossible since w e are taking M = a i and m ≥ p i m i / 2 ≥ m 9 / 10 i / 2, so m ≥ 2 µ − 2 log M , as required. W e ﬁnish the co nstruction of G by deﬁnin g wh ic h p airs of vertice s are adjacent. V ertices u, v ∈ V are adjacen t in G if there is i , 1 ≤ i ≤ s − 1, an edge ( X , Y ) of G i , and j ∈ { 1 , 2 } with u ∈ X j Y , v ∈ Y j X . An equiv alen t w ay of deﬁning the graph G is as follo ws . F or 1 ≤ j < i , let G j,i denote th e graph with v ertex set P i , where X, Y ∈ P i is an edge of G j,i if there are X ′ , Y ′ ∈ P j that are adjacen t in G j , and d ∈ { 1 , 2 } w ith X ⊂ X ′ d Y ′ and Y ⊂ Y ′ d X ′ . F or 1 < i ≤ s , let G i denote th e graph on P i whose edge set is the union of the edge sets of G 1 ,i , . . . , G i − 1 ,i . Finally , tw o v ertices u, v ∈ V are adjacen t in G if there is an edge ( X, Y ) of G s with u ∈ X and v ∈ Y . Note that G 1 is simply the empt y graph on P 1 . W e say that a sub set Z β -overlaps another set X if | X ∩ Z | ≥ β | Z | , that is, if a β -fraction of Z is in X . A set Z is β -c ontaine d in a partition P of V if there is a set X ∈ P such that Z β -ov erlaps X . An equitable p artition Z of V is a ( β , υ ) -r eﬁnement of a partition P of V if, f or at least (1 − υ ) |Z | sets Z ∈ Z , the set Z is (1 − β )-con tained in P . I n particular, when β = υ = 0, this notion agrees with the s tandard n otion of reﬁnement. That is, Z is a reﬁnemen t of P is equiv alent to Z b eing a (0 , 0)-reﬁnement of P . Our main resu lt, from wh ic h Theorem 1.1 easily f ollo ws, no w sa ys th at for an appropriate c h oice of p i , every regular partition of G must b e close to a reﬁnement of P s − 1 . In the pro of of Th eorem 1.1, Theorem 3.1 will b e used only in the ca se a = s − 1. Ho wev er, f or the lo wer b ound on the strong regularit y lemma in Theorem 1.2, we will need to apply T heorem 3.1 f or v arious v alues of a . This is wh y the parameter a is in tro du ced. Theorem 3.1 L et ν = 3 P s − 1 i =1 p i , and supp ose p i > 2 10 η m 2 1 for 1 ≤ i ≤ a , 1 − 2 7 ν > ǫ , β = 20 m − 3 / 2 1 , δ < β / 4 , and υ = 5 m − 1 / 2 1 . Wi th p ositive pr ob ability, th e r andom gr aph G has the fol lowing pr op e rty. Every ( ǫ, δ, η ) -r e gular e qui table p artition of G is a ( β , υ ) -r eﬁnement of P a . 3.2 Edge distr ibution in G Ha ving deﬁned th e (random) graph G , w e now show that w ith p ositiv e probabilit y G sat isﬁes certain prop erties (see Lemma 3.11) concerning its edge distribution whic h we will use to pr o ve Theorem 1.1. Note that G is determined by the G i . F or some of the d esired pr op erties, it will b e enough to sho w 17 that the edges in eac h G i are su ﬃcien tly uniform. F or other pr op erties, w e will need to consider ho w the edge distribution b etw een the v arious G i in teract with eac h other. In b ounding the p robabilities of certain ev en ts, w e will often consider the pr obabilit y of the ev ent giv en G i is pic ke d at random conditioned on the ev en t that G j with j < i are already c hosen. In the random graph G ( n, p ) on n v ertices w ith eac h edge tak en with probability p ind ep endent ly of the other edges, the exp ected degree of eac h ve rtex is p ( n − 1), and the follo wing sim p le lemma sh o ws that with high pr obabilit y no vertex will hav e degree which deviates muc h f rom this quan tit y . W e will assume through ou t this sub s ection th at n ≥ m 1 ≥ 2 200 . Lemma 3.1 The pr ob ability that in the r andom gr aph G ( n, p ) ther e is a vertex v whose de gr e e satisﬁes | de g ( v ) − pn | > n 3 / 4 is at most e − n 1 / 2 . Pro of: F or a ﬁxed vertex v , its degree deg( v ) follo ws a bin omial distribu tion with parameters n − 1 and p . Note that if | deg( v ) − pn | > n 3 / 4 then also | deg( v ) − p ( n − 1) | > n 3 / 4 − 1. F r om the Chernoﬀ-t yp e estimate (1), we get th at the probabilit y | deg( v ) − pn | > n 3 / 4 is at most 2 e − 2( n 3 / 4 − 1) 2 / ( n − 1) ≤ 1 n e − n 1 / 2 . As there are n vertic es, from the union b ound, w e get the pr obabilit y that there is a v ertex v w ith | deg( v ) − pn | > n 3 / 4 is at m ost e − n 1 / 2 . ✷ F or X ∈ P i , we w ill use N ( X ) to denote the neigh b orh o od of X in grap h G i , that is, the set of Y ∈ P i suc h that ( X, Y ) is an edge of G i . W e ha v e the follo wing corollary of Lemma 3.1. Corollary 3.1 L et E 1 b e the event that ther e is i , 1 ≤ i ≤ s − 1 , suc h that G i has a vertex X with de gr e e | N ( X ) | satisfying || N ( X ) | − p i m i | > m 3 / 4 i . The pr ob ability of event E 1 is at most π 1 := P s − 1 i =1 e − m 1 / 2 i . Lemma 3.2 Supp ose ν = 3 P s − 1 i =1 p i ≤ 1 / 2 . F or 2 ≤ i ≤ s − 1 , let E 2 i b e the event tha t G i has less than 1 4 p i m 2 i e dges which ar e not e dges of G i . L et E 2 b e the event that none of the events E 2 i , 2 ≤ i ≤ s − 1 , o c curs. The pr ob ability π 2 of event E 2 is at most π 1 + P s − 1 i =2 e − p 2 i m 2 i / 24 , wher e π 1 is deﬁne d in Cor ol lary 3.1. Pro of: If ev en t E 1 do es not occur, giv en ν ≤ 1 / 2, then the n umb er of edges of G i is at m ost   i − 1 X j = 1 p j + m − 1 / 4 j   m 2 i / 2 ≤ ν 4 m 2 i ≤ m 2 i / 8 . Eac h of th e remainin g at least  m i 2  − 1 8 m 2 i ≥ m 2 i / 3 unord ered pairs of parts of P i has probabilt y p i of b eing an edge of G i , ind ep endently of eac h other. The exp ected num b er of edges of G i whic h are n ot edges of G i is th erefore at least p i m 2 i 3 = p i m 2 i 4 + p i m 2 i 12 . By (1), the pr ob ab ility of ev en t E 2 i giv en the n umber of edges of G i is at most m 2 i / 8 is at m ost e − 2( p i m 2 i / 12) 2 / ( m 2 i / 3) = e − p 2 i m 2 i / 24 . Summing o v er all i , the probabilit y of ev ent E 2 giv en E 1 do es not occur is at most P s − 1 i =2 e − p 2 i m 2 i / 24 . W e th us ha ve that the p robabilit y of E 2 is at most π 2 . ✷ 18 In a graph G w ith v ertex subsets U, W , w e let d G ( U, W ) den ote the fraction of p airs in U × W wh ich are edges of G . If U = { u } consists of a single verte x u , we let d G ( u, W ) = d G ( U, W ). If th e underlying graph G is clear, we will sometimes write d ( U, W ) for d G ( U, W ). The follo w in g lemma sho ws that there is a lo w probabilit y that the density b et ween a vertex and certain v ertex sub sets is large. Lemma 3.3 L et E 3 b e the event that ther e is i , 1 ≤ i ≤ s − 2 , and distinct X , Y ∈ P i , d ∈ { 1 , 2 } , and v ∈ X 3 − d Y , such that ( X , Y ) is an e dge of G i but not an e dge of G i , and d G ( v , Y d X ) > ν . The pr ob ability of event E 3 is at mo st π 3 := P s − 2 i =1 P s − 1 j = i +1 m i m j e − 4 p 2 j m j /m i . Pro of: If ( X, Y ) is an edge of G i but not an edge of G i , then none of the edges of G b et w een X 3 − d Y and Y d X come from the edges of an y G j with j ≤ i . So f or ev en t E 3 to o ccur, there must b e 1 ≤ i < j ≤ s − 1, X , Y ∈ P i with ( X, Y ) an edge of G i , and X ′ ∈ P j with X ′ ⊂ X 3 − d Y , such that d G j ( X ′ , Y ∗ ) > 3 p j , wh ere Y ∗ denotes the set of Y ′ ∈ P j with Y ′ ⊂ Y d X . Fix for no w i , 1 ≤ i ≤ s − 2, and j with i + 1 ≤ j ≤ s − 1. Fix also an edge ( X, Y ) of G i whic h is not an edge of G i and d ∈ { 1 , 2 } . Fix a set X ′ ∈ P j with X ′ ⊂ X 3 − d Y and as b efore let Y ∗ denote the s et of all Y ′ ∈ P j with Y ′ ⊂ Y d X . The pr ob ab ility that d G j ( X ′ , Y ∗ ) > 3 p j is b y (1) at most e − 2(2 p j | Y ∗ | ) 2 / | Y ∗ | = e − 4 p 2 j m j /m i , since | Y ∗ | = m j 2 m i . Sum m ing o ver all p ossible choi ces of i , j , ( X, Y ), d , and X ′ ∈ P j with X ′ ⊂ X 3 − d Y , by the u nion b ound we ha v e the pr obabilit y of ev en t E 3 is at m ost s − 2 X i =1 s − 1 X j = i +1 2 m 2 i · m j 2 m i e − 4 p 2 j m j /m i = π 3 . ✷ Note that the condition that ( X, Y ) is an edge of G i but n ot of G i is n ecessary , since it guaran tees that none of the edges in G j with j ≤ i con trib u tes to the edges b et w een X 3 − d Y and Y d X in G . If ( X, Y ) wa s an edge of G i , then we w ould ha v e a complete bipartite graph b et w een X and Y and h ence d G ( v , Y d X ) = 1. The cod egree codeg( u, v ) of t w o v ertices u and v is the num b er of vertice s w whic h are co nnected to b oth u and v . A second u seful fact ab ou t G ( n, p ) is that with high probability the co degree of an y t wo vertic es u and v is roughly p 2 n . Lemma 3.4 The pr ob ability that in the r andom gr aph G ( n, p ) ther e ar e distinct ve rtic e s u and v with | c o de g ( u, v ) − p 2 n | > n 3 / 4 is at most e − n 1 / 2 . Pro of: F or ﬁxed d istinct ve rtices u and v , the co degree cod eg( u, v ) is binomially d istr ibuted with parameters n − 2 and p 2 . Note that if | cod eg ( u, v ) − p 2 n | > n 3 / 4 then | cod eg ( u, v ) − p 2 ( n − 2) | > n 3 / 4 − 2. By Chernoﬀ ’s inequalit y (1), the probabilit y that | co deg( u, v ) − p 2 n | > n 3 / 4 is at most 2 e − 2( n 3 / 4 − 2) 2 / ( n − 2) ≤ n − 2 e − n 1 / 2 . Usin g the union b ound o ver all  n 2  c hoices of u and v yields the result. ✷ W e ha v e the follo wing corolla ry of Lemma 3.4. 19 Corollary 3.2 L et E 4 b e the eve nt th at ther e is i , 1 ≤ i ≤ s − 1 , such that G i has vertic es X , Y ∈ P i with c o de gr e e satisfying | c o de g ( X , Y ) − p 2 i m i | > m 3 / 4 i . The pr ob ability of event E 4 is at most π 4 = P s − 1 i =1 e − m 1 / 2 i . F or X ∈ P i , let U ( X ) = S Y ∈ N ( X ) Y . Th e follo wing three lemmas will b e used to prov e Lemma 3.8, whic h b oun d s the pr obabilit y th at there is i , 1 ≤ i ≤ s − 1, X ∈ P i , and a vertex v 6∈ X su c h that d G ( v , U ( X )) > ν . The pro of, whic h puts together the next three lemmas, m ak es sure that it is unlik ely that any G j con trib u tes to o m uch to the densit y b et ween v and U ( X ). Lemma 3.5 Fix 1 ≤ i < s . The pr ob ability th at ther e is a p air of distinct sets X , Y ∈ P i which satisfy d G i ( Y , N ( X )) > 2 p i is at most π 5 i := 2 e − m 1 / 2 i . Pro of: F rom Lemma 3.1, we know that | N ( X ) | ≥ p i m i − m 3 / 4 i ≥ 3 p i m i / 4 for all X ∈ P i with probabilit y at least 1 − e − m 1 / 2 i . Also, b y Lemma 3.4, we ha ve co deg( X , Y ) ≤ p 2 i m i + m 3 / 4 i ≤ 3 p 2 i m i / 2 for all distinct X, Y ∈ P i in graph G i with pr obabilit y at least 1 − e − m 1 / 2 i . The num b er of edges b et w een Y and N ( X ) in G i is just the co degree co deg( X , Y ) of X and Y in G i . Therefore, d G i ( Y , N ( X )) = co deg( X, Y ) / | N ( X ) | . Hence, with probabilit y at least 1 − 2 e − m 1 / 2 i , we h a ve d G i ( Y , N ( X )) = co deg( X, Y ) | N ( X ) | ≤ 3 p 2 i m i / 2 3 p i m i / 4 = 2 p i . ✷ Lemma 3.6 Fix 1 < i < s . Supp ose every vertex of G i has de gr e e at most ν i m i / 2 , wher e ν i = 3 P j ν i is at most π ′ 5 i := e − m 1 / 2 i + m 2 i e − ν i p 2 i m i / 4 . Pro of: Note that G i is determined b y G 1 , . . . , G i − 1 . W e sh o w that, conditioning on G i has maximum degree at most ν i m i / 2, the r an d om graph G i is suc h that the probabilit y that th ere are distinct sets X, Y ∈ P i whic h satisfy d G i ( Y , N ( X )) > ν i is at most e − m 1 / 2 i + m 2 i e − ν i p 2 i m i / 4 . Fix for n o w X , Y ∈ P i . Let U b e the n eigh b orho o d of Y in G i . By assum p tion | U | ≤ ν i m i / 2. The exp ectation of | N ( X ) ∩ U | is at most p i ν i m i / 2. The probabilit y that | N ( X ) ∩ U | > 3 p i ν i m i / 4 is b y (1) at most e − 2( p i ν i m i / 4) 2 / | U | ≤ e − 2( ν i p i m i / 4) 2 / ( ν i m i / 2) = e − ν i p 2 i m i / 4 . By Lemma 3.1, w e kno w that | N ( X ) | ≥ 3 p i m i / 4 for al l X ∈ P i with probabilit y at least 1 − e − m 1 / 2 i . Therefore, using the union b ou n d, w ith probabilit y at least 1 − e − m 1 / 2 i − m 2 i e − ν i p 2 i m i / 4 , w e hav e d G i ( Y , N ( X )) = | N ( X ) ∩ U | | N ( X ) | ≤ (3 ν i p i m i / 4) (3 p i m i / 4) = ν i , for all X, Y ∈ P i . ✷ 20 Lemma 3.7 Fix 1 ≤ i < j < s . Supp ose every vertex of G i has de gr e e at le ast p i m i / 2 . L et E b e the event that ther e is a set X ∈ P i and a set Y ∈ P j with Y 6⊂ X with mor e than 2 p j | N ( X ) | m j m i neighb ors Y ′ in G j with Y ′ ⊂ U ( X ) . The pr ob ability of event E is at most π 5 ij := m i m j e − p 2 j p i m j . Pro of: The num b er of Y ′ ∈ P j with Y ′ ⊂ U ( X ) is | N ( X ) | m j m i . The probabilit y that a giv en Y has at least 2 p j | N ( X ) | m j m i neigh b ors Y ′ in G j with Y ′ ⊂ U ( X ) is, b y (1), at most e − 2  p j | N ( X ) | m j m i  2 /  | N ( X ) | m j m i  = e − 2 p 2 j | N ( X ) | m j m i . As there are at most m j m i suc h pairs X, Y , we h a ve by the union b ound, the probabilit y of ev en t E is at most m i m j e − 2 p 2 j | N ( X ) | m j m i ≤ m i m j e − p 2 j p i m j . ✷ F r om the previous three lemmas, w e get the n ext lemma. Lemma 3.8 Consider the event E 5 that ther e is i , 1 ≤ i ≤ s − 1 , X ∈ P i , and vertex v 6∈ X with d G ( v , U ( X )) > ν . The pr ob ability of eve nt E 5 is at most π 5 := π 1 + P s − 1 i =1 π 5 i + P s − 1 i =2 π ′ 5 i + P 1 ≤ i i . F or even t E 5 to o ccur at least one of the follo wing ev ents o ccurs : • There is 1 ≤ i < s and distinct sets X , Y ∈ P i with d G i ( Y , N ( X )) > 3 p i . • There is 1 ν i = P j 3 p j . The ﬁrst case is co v ered by Lemma 3.5, the second b y Lemma 3.6 and the third b y Lemma 3.7. F or Lemmas 3.6 and 3.7 to b e applicable, it is enough to know also that for any i and an y X ∈ P i , | N ( X ) − p i m i | ≤ m 3 / 4 i ≤ p i m i / 4. But this is j ust the ev ent that E 1 do es not occur . F rom Corollary 3.1, we kno w this holds with pr obabilit y at least 1 − π 1 . Therefore, putting everything together, the probabilit y of eve nt E 5 is at m ost π 1 + s − 1 X i =1 π 5 i + s − 1 X i =2 π ′ 5 i + X 1 ≤ i 5 √ h, (7) wher e h = h ( u 1 , u 2 ) = u 1 u 2 2 ln m i e u 2 . The pr ob ability of event E 7 is at most π 7 := P s − 1 i =1 2 m − 2 i . Pro of: By L emm a 2.6, f or eac h i , the pr obabilit y that there are subsets U 1 , U 2 ⊂ P i suc h that (7) fails is at most 2 m − 2 i . By the union b ound, the probabilit y of ev ent E 7 is at m ost P s − 1 i =1 2 m − 2 i . ✷ W e gather the p revious lemmas into one r esult, which sho ws that w ith p ositiv e probability the edge distribution o f G h as certa in desirable prop erties. Lemma 3.11 Supp ose ν = 3 P s − 1 i =1 p i ≤ 1 / 2 . With pr ob ability at le ast 1 / 2 , the gr aph G has the fol lowing pr op erties for al l i , 1 ≤ i ≤ s − 1 . • The de g r e e of every vertex in gr aph G i diﬀers fr om p i m i by at most m 3 / 4 i and the c o de gr e e of every p air of distinct vertic es diﬀers fr om p 2 i m i by at most m 3 / 4 i . • The numb er of e dges of G i not in G i is at le ast p i m 2 i / 4 . • F or al l X ∈ P i and vertex v 6∈ X , we have d G ( v , U ( X )) ≤ ν . • F or al l distinct X , Y ∈ P i , d ∈ { 1 , 2 } , and v ∈ X 3 − d Y , such that ( X, Y ) is an e dge of G i but X and Y ar e not adjac ent in G i , we have d G ( v , Y d X ) ≤ ν . • F or al l X ∈ P i , X has at most ν | N ( X ) | neighb ors Y in G i with Y ∈ N ( X ) . 22 • F or al l vertex subsets U 1 , U 2 ⊂ P i of gr aph G i with | U 1 | = u 1 , | U 2 | = u 2 , and u 1 ≤ u 2 , | e ( U 1 , U 2 ) − p i u 1 u 2 | ≤ 5 √ h, wher e h = h ( u 1 , u 2 ) = u 1 u 2 2 ln m i e u 2 . Pro of: By Corollaries 3.1, 3.2 and L emm as 3.2, 3.3, 3.8, 3.9, 3.10 and the union b ound, the probabilit y that at least one E h , 1 ≤ h ≤ 7, o ccurs is at most P 7 h =1 Pr[ E h ] ≤ P 7 h =1 π h . Using the estimat es ρ = 2 − 20 , m 1 ≥ 2 200 , m r = m r − 1 a r − 1 for 2 ≤ r ≤ s , where a r − 1 = 2 ⌊ ρm 9 / 10 r − 1 ⌋ , and p i ≥ m − 1 / 10 i for 1 ≤ r ≤ s − 1, it is easy to verify that eac h π h ≤ 1 / 14 and h ence the probabilit y that n one of these ev ents o ccur, i.e., G h as the desired prop erties, is at least 1 / 2. ✷ F or the rest of the pro of of Theorem 3.1, w e su p p ose that G has th e p rop erties describ ed in Lemma 3.11. 3.3 Regular partitions are close to b eing reﬁnemen ts Let θ = m − 1 / 2 1 , ζ = ω = 20 θ , β = ζ m 1 , and γ = 1 − ω . Supp ose for co nt radiction that there is an equitable partition Z : V = Z 1 ∪ . . . ∪ Z k of the vertex set of G suc h that all but at most ηk 2 ordered pairs ( Z j , Z ℓ ) of parts are ( ǫ, δ )-regular, but Z is n ot a ( β , υ )-reﬁnement of P a . The t wo main lemmas for the pro of are Lemmas 3.14 and 3.15, which show that if Z j satisﬁes certain conditions, then there are at least θ − 1 η k pairs ( Z j , Z ℓ ) that are not ( ǫ, δ )-regular. The rest of th e pro of, Theorem 3.2, shows that there are at lea st θ k Z j whic h sati sfy the conditions of Lemmas 3.1 4 or 3.15. T ogether, w e ge t at least θ − 1 η k · θ k = η k 2 ordered pairs ( Z j , Z ℓ ) whic h are not ( ǫ, δ )-regular, whic h completes the pr o of. Since P 1 is a partition into m 1 parts, then, by the p igeonhole principle, eac h Z j is 1 m 1 -con tained in P 1 . W e call Z j rip e with resp ect to r if Z j is β -con tained in P r but n ot (1 − β )-con tained in P r . That is, Z j is r ip e if there is X ∈ P r con taining a β -fraction of it bu t no X ∈ P r con taining a (1 − β )-fraction of it. Let ψ = 2 20 β . W e call Z j shatter e d with resp ect to r if Z j is (1 − β )-con tained in P r , but at least a ψ -fraction of Z j is cont ained in sub s ets X ∩ Z j with X ∈ P r +1 and | X ∩ Z j | < β | Z j | . The sense here is that Z j is shattered by th e partition P r +1 if Z j is almost completely con tained in some X ∈ P r but it is not w ell-co v er ed by P r +1 . W e sa y that a s u bset X ⊂ V ( β , γ ) -supp orts the p artition Z if at least a γ -fraction of th e element s of X are in sets Z j whic h β -o verla p X . That is, a γ -fraction of the elemen ts of X are in sets Z j for whic h | X ∩ Z j | ≥ β | Z j | . Lemma 3.12 Each of the m 1 sets in the p artition P 1 ( β , 1 − β m 1 ) -supp orts Z . Pro of: Let X ∈ P 1 . A t most a β -fraction of V is in s ets o f the form X ∩ Z j with | X ∩ Z j | < β | Z j | . Hence, as | X | = | V | /m 1 , at most a β m 1 -fraction of X b elongs to Z i whic h do not β -o ve rlap X . ✷ Let S i denote the set of X ∈ P i whic h ( β , γ )-supp ort Z . W e will let κ i = | S i | | P i | . Let W i denote the set of X ∈ P i for which | N ( X ) ∩ S i | ≤ κ i | N ( X ) | / 4. 23 Lemma 3.13 F or 1 ≤ i ≤ s − 1 with κ i > 100 p − 2 i m − 1 i ln( m i e ) , we have | W i | ≤ 100 p − 2 i ln( κ − 1 i e ) . Pro of: In graph G i , the num b er e ( W i , S i ) of pairs in W i × S i whic h are edges is at most κ i / 4 times the sum of the degrees of the vertice s in W i . S ince, by Lemma 3.11, ev ery v ertex has degree at most 2 p i | P i | in G i , we h a ve e ( W i , S i ) ≤ | W i | · ( κ i / 4) · 2 p i | P i | = p i | S i || W i | / 2. Hence, b y Lemma 3.11, p i | S i || W i | / 2 ≤ | e ( W i , S i ) − p i | W i || S i || ≤ 5 √ h, where h = u 1 u 2 2 ln m i e u 2 , and u 1 = min( | W i | , | S i | ) and u 2 = max( | W i | , | S i | ). By sq u aring b oth sides, substituting u 1 u 2 = | W i || S i | and simplifying, we h a ve u 1 ≤ 100 p − 2 i ln m i e u 2 . I f u 1 = | S i | = κ i m i , then κ i m i = u 1 ≤ 100 p − 2 i ln m i e u 2 ≤ 100 p − 2 i ln( m i e ) , con tradicting our assum p tion. Hence, u 1 = | W i | , and | W i | ≤ 100 p − 2 i ln( κ − 1 i e ). ✷ The follo wing simple prop osition d emonstrates the hereditary nature of supp orting sets. Prop osition 3.1 Supp ose Y ∈ P i is such that Y ( β , γ ) -supp orts the p artition Z . Then, for e ach X ∈ P i distinct fr om Y and d ∈ { 1 , 2 } , Y d X ( β / 4 , 1 / 4) -supp orts the p artition Z . Pro of: W e will use th e fact γ ≥ 7 / 8. Th e sum of | Z t ∩ Y d X | ov er all Z t whic h β -o v erlap Y but do not β / 4-o v erlap Y d X is at most | Y | / 4. Sin ce Y ( β , γ )-supp orts the partition, the sum of | Z t ∩ Y d X | ov er all Z t whic h β / 4-o verla ps Y d X is at least | Y d X | − (1 − γ ) | Y | − | Y | / 4 ≥ | Y | / 8 = | Y d X | / 4 . Hence Y d X ( β / 4 , 1 / 4)-supp orts the partitio n Z . ✷ The follo wing lemma shows that if Z j satisﬁes certain conditions, then there are many (at least θ − 1 η k ) Z ℓ suc h that ( Z j , Z ℓ ) is not ( ǫ, δ )-regular. Lemma 3.14 Supp ose X ∈ P i \ W i , κ i = | S i | / | P i | ≥ 1 / 2 , Z j is shatt er e d with r esp e ct to i , and Z j (1 − β ) -overlaps X . Ther e ar e at le ast θ − 1 η k sets Z ℓ ∈ Z for which ( Z j , Z ℓ ) is not ( ǫ, δ ) -r e gu lar. Pro of: S in ce Z j is shattered with r esp ect to i and Z j (1 − β )-o verlaps X , then | X ∩ Z j | ≥ (1 − β ) | Z j | , but the sum of | X ′ ∩ Z j | ov er all X ′ ∈ P i +1 with | X ′ ∩ Z j | < β | Z j | is at least ψ | Z j | . Let Z ′ j = X ∩ Z j , so | Z ′ j | ≥ (1 − β ) | Z j | . F or eac h X ′ ∈ P i +1 with X ′ ⊂ X and | X ′ ∩ Z j | < β | Z j | , let λ X ′ = | X ′ ∩ Z ′ j | / | Z j | , so eac h λ X ′ < β , i.e., β > || λ || ∞ . Also, || λ || 1 ≥ ψ − β follo ws f rom the facts that | X ∩ Z j | ≥ (1 − β ) | Z j | and the sum of | X ′ ∩ Z j | ov er all X ′ ∈ P i +1 with | X ′ ∩ Z j | < β | Z j | is at least ψ | Z j | . T h erefore, σ 2 =  || λ || 2 || λ || 1  2 ≤ || λ || ∞ || λ || 1 < β ψ − β = 1 2 20 − 1 < 2 − 19 . 24 By Lemma 2.3 with α = 1 / 8, µ = 2 ρ 1 / 2 = 2 − 9 , σ < 2 − 9 , and τ = 1 − 2 − 5 , w e ha v e that the n umber of Y ∈ N ( X ) for w hic h | Z j ∩ X 1 Y | , | Z j ∩ X 2 Y | ≥ α || λ || 1 | Z j | ≥ α ( ψ − β ) | Z j | (8) is at least (1 − 2 − 5 ) | N ( X ) | , where N ( X ) is the neighb orh o o d of X in graph G i . By Lemm a 3.11, the num b er of Y ∈ N ( X ) wh ic h are also adjacent to X in G i is at most ν | N ( X ) | . Also, since X 6∈ W i , w e hav e | N ( X ) ∩ S i | ≥ κ i | N ( X ) | / 4. Therefore, the num b er of Y ∈ S i with ( X , Y ) an ed ge of G i but n ot an edge of G i , and (8) is satisﬁed is at least (1 − 2 − 5 ) | N ( X ) | − | N ( X ) \ S i | − ν | N ( X ) | > ( κ i / 4 − 2 − 5 − ν ) | N ( X ) | ≥ | N ( X ) | / 16 . Fix suc h a Y , and let U d = Z j ∩ X d Y for d ∈ { 1 , 2 } , so | U 1 | , | U 2 | ≥ α ( ψ − β ) | Z j | . Since Y ∈ S i , we ha ve Y ( β , γ )-supp orts Z . By Prop osition 3.1, Y d X ( β / 4 , 1 / 4)-supp orts Z . By Lemma 3.11, for eac h v ertex v ∈ X 3 − d Y , we hav e d ( v , Y d X ) ≤ ν . In particular, d ( U 3 − d , Y d X ) ≤ ν . Let R d b e the union of all Z ℓ ∩ Y d X suc h that Z ℓ β / 4-o v erlaps Y d X , so R d is a subset of Y d X of cardin alit y at least | Y d X | / 4. Hence, d ( U 3 − d , R d ) ≤ 4 ν . F or Z ℓ whic h β / 4-o v erlaps Y d X , let Z ′ ℓ = R d ∩ Z ℓ , so | Z ′ ℓ | ≥ β | Z ℓ | / 4. W e next show that there are many Z ′ ℓ whic h satisfy d ( U 3 − d , Z ′ ℓ ) ≤ 8 ν. (9) Indeed, the union of the Z ′ ℓ whic h do not satisfy (9) has cardinalit y at most 1 2 | R d | , so at least 1 / 2 of R d consists of the union of Z ′ ℓ whic h satisfy (9). Th e n umb er of ℓ whic h sati sfy (9 ) is at least 1 2 | R d | / | Z ℓ | ≥ 1 2 ( | Y d X | / 4) / | Z ℓ | = 1 16 | Y | / | Z ℓ | = 1 16 k /m i , where in the last equalit y w e used | Y | = | V | /m i and | Z ℓ | = | V | /k . F or eac h Z ′ ℓ whic h satisﬁes (9), we ha v e d ( U d , Z ′ ℓ ) = 1 since ( X, Y ) is an edge of G i and, therefore, the densit y of edges b et w een X d Y and Y d X is 1. Hence d ( U d , Z ′ ℓ ) − d ( U 3 − d , Z ′ ℓ ) ≥ 1 − 8 ν ≥ ǫ. Since also | U d | , | U 3 − d | ≥ α ( ψ − β ) | Z j | ≥ δ | Z j | , and | Z ′ ℓ | ≥ β 4 | Z ℓ | ≥ δ | Z ℓ | , w e ha ve in this case ( Z j , Z ℓ ) is n ot ( ǫ, δ )-regular. Since the n u mb er of suc h Y is at least | N ( X ) | / 16, we ha v e that the num b er of pairs ( Z ℓ , Y d X ) su c h that Z ℓ β / 4-o v erlaps Y d X and ( Z j , Z ℓ ) is not ( ǫ, δ )-regular is at least  1 16 k /m i  ( | N ( X ) | / 16) ≥ 2 − 9 p i k , where w e used | N ( X ) | ≥ 1 2 p i m i from Lemma 3.11. As Z ℓ β / 4-o v erlaps Y d X in ea c h such p air, a giv en Z ℓ is in at most 4 β − 1 suc h pairs. Hence, the num b er of Z ℓ for whic h ( Z j , Z ℓ ) is n ot ( ǫ, δ )-regular is at least 2 − 11 β p i k ≥ θ − 1 η k . ✷ Lik e Lemma 3.14, the next lemma sho ws that if Z j satisﬁes certain conditions, then th ere are at least θ − 1 η k Z ℓ suc h that ( Z j , Z ℓ ) is not ( ǫ, δ )-regular. 25 Lemma 3.15 Supp ose X ∈ P i \ W i , κ i ≥ 1 / 2 , Z j is rip e with r esp e ct to i , and Z j β -overlaps X . Then ther e ar e at le ast θ − 1 η k sets Z ℓ ∈ Z for which ( Z j , Z ℓ ) is not ( ǫ, δ ) -r e gular. Pro of: Since Z j is rip e with resp ect to i , | X ∩ Z j | < (1 − β ) | Z j | . T h erefore, letting U ′ = Z j \ X , w e ha ve | U ′ | ≥ β | Z j | . By L emm a 3.11, for eac h v ertex v of G wh ic h is not in X , w e h av e d ( v , U ( X )) ≤ ν . Sin ce X 6∈ W i , we ha ve | N ( X ) ∩ S i | ≥ κ i | N ( X ) | / 4 ≥ | N ( X ) | / 8 . (10) So d ( v , [ Y ∈ N ( X ) ∩ S i Y ) ≤ 8 ν . (11) Fix for this paragraph Y ∈ N ( X ) ∩ S i . S ince Z j β -o v erlaps X , there is d = d ( j, Y ) ∈ { 1 , 2 } suc h that Z j β / 2-o v erlaps X d Y . Let U Y = Z j ∩ X d Y , so | U Y | ≥ β 2 | Z j | and d ( U Y , Y d X ) = 1. As Y ∈ S i , w e ha ve Y ( β , γ )-sup p orts Z . By Prop osition 3.1, Y d X ( β / 4 , 1 / 4)-supp orts Z . F or Y ∈ N ( X ) ∩ S i , let R Y denote the s et of vertic es y wh ic h are in Y d X with d = d ( j, Y ), and y is also in a Z ℓ whic h β / 4-o ve rlaps Y d X , so | R Y | ≥ 1 4 | Y d X | = 1 8 | Y | = | V | 8 m i . (12) Let R = S Y ∈ N ( X ) ∩ S i R Y . W e ha v e | R | ≥ | N ( X ) ∩ S i | | V | 8 m i ≥ 2 − 6 | N ( X ) | | V | m i ≥ 2 − 6 p i m i 2 · | V | m i = 2 − 7 p i | V | , (13) where we used (12), (10), and | N ( X ) | ≥ p i m i / 2. By (11) and (12), we ha v e for v 6∈ X , d ( v , R ) ≤ 2 6 ν. (14) By (14), we ha ve d ( U ′ , R ) ≤ 2 6 ν . F or Z ℓ whic h β / 4-o verla ps Y d X for some Y ∈ N ( X ) ∩ S i and d = d ( j, Y ), let Z Y ℓ = Z ℓ ∩ Y d X , so | Z Y ℓ | ≥ ( β / 4) | Z ℓ | . By deﬁnition, for eac h Y ∈ N ( X ) ∩ S i , R Y is the union of the sets Z Y ℓ . W e next sho w that there are many Z Y ℓ whic h sati sfy d ( U ′ , Z Y ℓ ) ≤ 2 7 ν. (15) Indeed, the union of the Z Y ℓ whic h do not s atisfy (15) has cardinalit y at most 1 2 | R | , so at least 1 / 2 of R consists of the un ion of Z Y ℓ whic h satisfy (1 5). T he n umb er of pairs ( ℓ, Y ) w hic h satisfy (15) is at least 1 2 | R | / | Z ℓ | ≥ 1 2 2 − 7 p i | V | / | Z ℓ | = 2 − 8 p i k . 26 where we u sed (13) and | Z ℓ | = | V | /k . Since for ea c h suc h ℓ , w e ha v e Z ℓ β / 4-o v erlaps Y d X , eac h such ℓ is in at most 4 β − 1 of th e pairs ( ℓ, Y ) we just coun ted. Hence, the n umber of ℓ for which there is Y suc h that (15) holds is at least 2 − 10 β p i k . By (15) and d ( U Y , Z Y ℓ ) = 1, w e ha ve d ( U Y , Z Y ℓ ) − d ( U ′ , Z Y ℓ ) ≥ 1 − 2 7 ν > ǫ, and as | U Y | , | U ′ | ≥ β 2 | Z j | ≥ δ | Z j | and | Z Y ℓ | ≥ β 4 | Z ℓ | ≥ δ | Z ℓ | , we hav e that ( Z j , Z ℓ ) is not ( ǫ, δ )-regular for at least 2 − 10 β p i k ≥ θ − 1 η k v alues of ℓ . ✷ The follo wing theorem completes the pro of. Theorem 3.2 The numb er of or der e d p airs ( Z j , Z ℓ ) which ar e not ( ǫ, δ ) -r e gular is at le ast η k 2 . Pro of: By assumption, Z is n ot a ( β , υ )-reﬁn emen t of P a . Hence, th e num b er of parts Z j of p artition Z whic h are not (1 − β )-con tained in P a is at least υ k . Let i 0 b e the minim um p ositiv e inte ger for whic h P i 0 is not a ( β , υ )-reﬁnement of Z . As, by assumption, P a is not a ( β , υ )-reﬁnement of Z , we ha ve 1 ≤ i 0 ≤ a . Claim 3.1 We have κ 1 = 1 and κ i ≥ 1 / 2 for i < i 0 . As β = ζ /m 1 , by Lemma 3.12, eac h of the m 1 parts of partition P 1 ( β , 1 − ζ )-su p p orts Z j . As ζ = ω and γ = 1 − ω , it follo ws that S 1 = P 1 and κ 1 = | S 1 | / | P 1 | = 1. F r om the deﬁnition of i 0 , for eac h i < i 0 , P i 0 is a ( β , υ )-reﬁn emen t of Z . Fix for this paragraph suc h an i β and γ = 1 − ω , th e fr action of X ∈ P i whic h d o not ( β , γ )-sup p ort Z is at most β + υ ω . Hence κ i ≥ 1 − β + υ ω ≥ 1 / 2, wh ic h completes the pro of of Claim 3.1. Consider the partition Z = Z 1 ∪ Z 2 ∪ Z 3 ∪ Z 4 ∪ Z 5 ∪ Z 6 , wh ere Z j ∈ Z h if h is minim um suc h that Z j satisﬁes p rop ert y h b elo w. 1. T here is i < i 0 and X ∈ P i \ W i suc h that Z j is shattered with resp ect to i and (1 − β )-o verla ps X or if Z j is rip e with resp ect to i and β -o ve rlaps X , 2. F or ev ery X ∈ P 1 suc h that Z j β -o v erlaps X , X ∈ W 1 . 3. T here is i , 1 1 and Z j is rip e with resp ect to i 0 , and there is X ∈ W i 0 suc h that Z j β -o v erlaps X . 5. Z j is r ip e with resp ect to i 0 , and there is X ∈ P i 0 \ W i 0 suc h that Z j β -o v erlaps X . 6. Z j is (1 − β )-con tained in P i 0 . It is not immediately obvio us that the ab ov e six su bfamilies of Z form a partition of Z , so we ﬁr st sho w th at this is ind eed the ca se. 27 Claim 3.2 The ab ove six subfamilies form a p artition of Z . As Z j ∈ Z h if and only h is the minimum suc h that Z j satisﬁes prop erty h , the subfamilies Z h , 1 ≤ h ≤ 6, are pairwise disjoint. W e thus need to sho w that eac h Z j is in at least one Z h . S upp ose for contradict ion that Z j is in n one of the Z h . By prop ert y 6, Z j is not (1 − β )-con tained in P i 0 . If Z j is β -con tained in P i 0 , then Z j is rip e with resp ect to i 0 , and there is X ∈ P i 0 suc h that Z j β -o v erlaps X . Either ev ery suc h X ∈ W i 0 or there is such an X 6∈ W i 0 , and by prop erties 2, 4 an d 5, we m u st ha ve in this case Z j is in a Z h for some h ≤ 5. So Z j is not β -con tained in P i 0 , and n oting that ev ery Z j is β -conta ined in P 1 , we must ha v e Z j is rip e or shattered w ith resp ect to at least one i with 1 ≤ i < i 0 . In particular, there is i < i 0 and X ∈ P i suc h that Z j is shattered with resp ect to i and (1 − β )-o v erlaps X or Z j is rip e with r esp ect to i and β -o v erlaps X . Sin ce Z j 6∈ Z 1 , for every su c h i < i 0 and X ∈ P i , we m ust h a ve X ∈ W i . Bu t then Z j ∈ Z 2 or Z 3 . T h us Z j is in at least on e of the six s u bfamilies, completing the claim that that these subfamilies in deed form a partition of Z . As the n u m b er of p arts Z j of p artition Z whic h are not (1 − β )-con tained in P i 0 is at least υ k , we hav e |Z \ Z 6 | ≥ υ k . (16) Let w i = | W i | / | P i | . By Clai m 3.1, κ 1 = 1 and κ i ≥ 1 / 2 for i < i 0 . Hence, from Lemma 3.13, we ha ve w 1 ≤ 100 p − 2 1 m − 1 1 ln(2 e ) ≤ m − 1 / 2 1 and similarly w := P 1 1. In ord er to giv e a low er b ound on κ i 0 , we next giv e an upp er b ound on the union of all sets Z j ∩ X with | Z j ∩ X | < β | Z j | an d X ∈ P i 0 . If Z j is not (1 − β )-con tained in P i 0 , then it m ust b e shattered or rip e with resp ect to some i w ith i < i 0 , or must hav e at m ost ψ | Z j | vertices in parts X ∩ Z j with X ∈ P i 0 and | X ∩ Z j | < β | Z j | . Eac h Z j whic h is sh attered or rip e with r esp ect to some i with i < i 0 is in Z 1 , Z 2 , or Z 3 , and hence the num b er of such Z j is at most |Z 1 ∪ Z 2 ∪ Z 3 | ≤ θ k + (2 m − 1 / 2 1 ) k + β − 1 m − 1 / 2 2 k . (17) Ev ery set Z j whic h (1 − β )-o verlaps P i 0 has at most a β -fraction of it con tained in sets X ∩ Z j with | X ∩ Z j | < β | Z j | and X ∈ P i 0 . In total, we get that the f r action of v ertices which b elong to one of the 28 sets X ∩ Z j with | X ∩ Z j | < β | Z j | and X ∈ P i 0 is at m ost θ + (2 m − 1 / 2 1 ) + β − 1 m − 1 / 2 2 + β + ψ . The fr action of sets in P i 0 whic h do not ( β , γ )-sup p ort Z is therefore 1 − κ i 0 ≤ ω − 1  θ + (2 m − 1 / 2 1 ) + β − 1 m − 1 / 2 2 + β + ψ  ≤ 1 / 2 . Hence, κ i 0 ≥ 1 / 2, whic h co mpletes Claim 3.3. Noting th at κ i 0 ≥ 1 / 2, the same argument that b ounded |Z 3 | also giv es that |Z 4 | ≤ β − 1 m − 1 / 2 2 k . (18) F r om the b ounds (16), (17), (18), we h a ve |Z 5 | ≥ |Z \ Z 6 | − |Z 1 ∪ Z 2 ∪ Z 3 | − |Z 4 | ≥ υ k −  θ k + 2 m − 1 / 2 1 k + β − 1 m − 1 / 2 2 k  − β − 1 m − 1 / 2 2 k ≥ θ k . As κ i 0 ≥ 1 / 2, by Lemma 3.15, eac h Z j ∈ Z 5 is in at least θ − 1 η k p airs ( Z j , Z ℓ ) w h ic h are not ( ǫ, δ )- regular. Hence, the num b er of irregular pairs is at least |Z 5 | θ − 1 η k ≥ η k 2 , whic h completes the pr o of. ✷ 3.4 Pro of of Theorem 1.1 T o prov e T heorem 1.1, it suﬃces to pro ve th e follo wing theorem. Corollary 3.3 L et ǫ = 1 / 2 , δ = 2 − 400 , η < 2 − 700 , s = ⌊ 2 − 600 η − 1 ⌋ , and k b e at most a tower of twos of height s . Ther e is a gr aph G = ( V , E ) for which any e qui table p artition Z of V into at most k p arts has at le ast η k 2 or der e d p airs of p arts which ar e not ( ǫ, δ ) -r e gular. Pro of: Let m 1 = 2 200 and p i = max( m − 1 / 10 i , 2 500 η ) for 1 ≤ i ≤ s − 1 and consider the graph G given with p ositiv e probabilit y b y Theorem 3. 1. As ν = 3 P s − 1 i =1 p i , we h a ve ν ≤ 3 s − 1 X i =1 ( m − 1 / 10 i + 2 500 η ) = 3 · 2 500 η ( s − 1) + 3 s − 1 X i =1 m − 1 / 10 i ≤ 3 · 2 − 100 + 3 3 2 p 1 < 6 p 1 < 2 − 10 , so 1 − 2 7 ν > ǫ . The ﬁrst inequ alit y uses that the maxim um of t w o nonnegativ e real n um b ers is at most their sum. The second inequalit y u ses s = ⌊ 2 − 600 η − 1 ⌋ and the fact that the su m of m − 1 / 10 i rapidly con verges, and p 1 = m − 1 / 10 1 = 2 − 20 . Note that as m 1 = 2 200 > 2 2 2 2 and m i ≥ 2 m 1 / 2 i − 1 for i > 1, we ha v e | P i | = m i is greater than a tow er of t w os of heigh t i + 2 for 1 ≤ i ≤ s . By Theorem 3.1 with a = s − 1, any ( ǫ, δ, η )-regular equitable partition of G is a ( β , υ )-reﬁn emen t of P s − 1 . I n particular, at lea st one part of P s − 1 con tains at least a (1 − β )-fraction of a part fr om Z . As 1 − β > 1 / 2, th is implies |Z | ≥ 1 2 | P s − 1 | > k , whic h completes the p ro of. ✷ 29 4 Lo w er b ound for the strong regularit y lemma In this secti on w e pro ve T heorem 1.2, whic h g iv es a low er b ound on the strong regularit y lemma an d states the follo wing. Let 0 < ǫ < 2 − 100 and f : N → (0 , 1) b e a decreasing function w ith f (1) ≤ 2 − 100 ǫ 6 . Deﬁne W ℓ recursiv ely by W 1 = 1, W ℓ +1 = T  2 − 70 ǫ 5 /f ( W ℓ )  , w here T is the to wer fu n ction deﬁned in the introd u ction. Let W = W t − 1 with t = 2 − 20 ǫ − 1 . Then there is a graph G such th at if equitable partitions A , B of the vertex set of G satisfy q ( B ) ≤ q ( A ) + ǫ and B is f ( |A| )-regular, then |A| , |B | ≥ W . W e next describ e th e pro of of Th eorem 1.2. In the ﬁ rst subsection, w e co nstruct the graph G as a sp ecializatio n of the construction in Theorem 3.1. The graph G w e us e to p ro ve Theorem 1.2 h as v ertex partitio ns P i,j with 1 ≤ i ≤ t , and 1 ≤ j ≤ h i satisfying P i ′ ,j ′ is a reﬁn emen t of P i,j if i ′ = i and j ′ > j or if i ′ > i . F urthermore, as the num b er of parts in eac h successive reﬁnemen t is rou gh ly exp onent ial in the num b er of parts in the previous partition, we sho w in the ﬁrst su bsection that | P t − 2 ,h t − 2 − 2 | ≥ W . Th e edges of G are deﬁned based on certain graphs G i,j on P i,j . In Sub section 4.3, w e pr o ve a lemma whic h implies that th e construction h as the prop ert y that q ( P i,h i ) > q ( P i,h i − 2 ) + 2 ǫ (19) for eac h i < t . Let A an d B b e equitable partitions of the v ertex set of G suc h th at q ( B ) ≤ q ( A ) + ǫ and B is f ( |A| )- regular. Let M 1 = 1 and M ℓ = | P ℓ − 1 ,h ℓ − 1 − 2 | for 1 < ℓ ≤ t − 1. Let r with 1 ≤ r ≤ t − 1 b e maxim um suc h that |A | ≥ M r . Let P ′ = P r,h r − 2 and P = P r,h r . In Su bsection 4.1, after deﬁnin g G , we sho w that it satisﬁes the h yp othesis of Th eorem 3.1, an d conclude that, as B is an f ( |A| )-regular partitio n of G and f is a decreasing function, it m ust b e close to b eing a reﬁn emen t of P . I t follo ws that if |A| ≥ M t − 1 = | P t − 2 ,h t − 2 − 2 | > W , then |B | > W as w ell, and we are done in this case. Thus we ma y assume |A| < M t − 1 and hen ce r ≤ t − 2. In Su bsection 4.4 we pro v e q ( A ) < q ( P ′ ) + ǫ 2 . (20) This f ollo ws from a lemma that states that q ( P ′ ) is close to th e maxim um m ean square den s it y density o ver all partitions of the same num b er of parts as P ′ . I n Subs ection 4.2, w e use the result th at B is close to b eing a r eﬁnemen t of P to conclude q ( P ) ≤ q ( B ) + ǫ 2 . (21) Putting the three estimates (19) (with i = r and n oting in this case P i,h i = P , P i,h i − 2 = P ′ ), (20 ), (21) together, w e get that q ( B ) ≥ q ( P ) − ǫ 2 > q ( P ′ ) + 2 ǫ − ǫ 2 > q ( A ) + ǫ, con tradicting the hyp othesis of Theorem 1.2, and completing the pro of of Theorem 1.2. ✷ 4.1 Construction of G and pro o f that B is an app ro ximate reﬁnemen t W e will co nstruct the graph G as a sp ecial case of the construction in Theorem 3.1. 30 Let t = 2 − 20 ǫ − 1 . W e ha v e p artitions P ℓ,j of the v ertex set V for 1 ≤ ℓ ≤ t and 1 ≤ j ≤ h ℓ , where h ℓ is d eﬁned later in the paragraph and P ℓ,j = P i are the partitio ns used to construct G in Theorem 3.1 with i = j + P d<ℓ h d . W e set m ℓ,j = | P ℓ,j | = | P i | = m i , and p ℓ,j = p i . As abov e, let M 1 = 1 and M ℓ = m ℓ − 1 ,h ℓ − 1 − 2 for 1 < ℓ ≤ t . Let ǫ ℓ = f ( M ℓ ), h ℓ = ǫ 5 2 70 ǫ ℓ , and p ℓ,j = max( m − 1 / 10 ℓ,j , 2 30 ǫ − 4 ǫ ℓ ) for 1 ≤ j ≤ h ℓ with j 6 = h ℓ − 1, and p ℓ,h ℓ − 1 = max( m − 1 / 10 ℓ,j , 2 30 ǫ − 4 ǫ ℓ , 2 10 ǫ ). Let m 1 = 2 10 ǫ − 2 , so m 1 ≥ 2 200 . Note that, as eac h m ℓ,j is exp onent ial in a p o wer of m ℓ,j − 1 , we get that M ℓ is at least a to wer of 2s of height h ℓ . That is, M ℓ ≥ T  2 − 70 ǫ 5 /f ( M ℓ − 1 )  . In particular, by induction, M ℓ ≥ W ℓ , where W ℓ is d eﬁned earlier in this section. W e will apply Theorem 3.1 to conclude the follo wing corollary whic h states that an y suﬃ ciently regular partition of G is roughly a reﬁnement of a particular P ℓ,j . T o accomplish this we need to sho w th at the conditions of the th eorem hold, whic h w e p ostp one until after stating the follo wing corolla ry . W e ﬁx G to be a graph satisfying the prop erties of Lemma 3. 11 so that if G also satisﬁes the conditions stated in T heorem 3.1, then it satisﬁes th e conclusion of T heorem 3.1. Corollary 4.1 L et r ≤ t − 1 b e the maximum p ositive inte ger for which |A| ≥ M r , so f ( |A| ) ≤ f ( M r ) = ǫ r , and P = P r,h r . The p artition B , which is ǫ r -r e gular, is a ( β , υ ) -r eﬁnement of P with β = 20 m − 3 / 2 1 and υ = 5 m − 1 / 2 1 . Note that ν = 3 s − 1 X i =1 p i = t X ℓ =1 h ℓ X j = 1 p ℓ,j ≤ 2 10 ǫt + t X ℓ =1 h ℓ X j = 1  m − 1 / 10 ℓ,j + 2 30 ǫ − 4 ǫ ℓ  ≤ 2 10 ǫt + s − 1 X i =1 m − 1 / 10 i + t X ℓ =1 h ℓ X j = 1 2 30 ǫ − 4 ǫ ℓ ≤ 2 10 ǫt + s − 1 X i =1 m − 1 / 10 i + t X ℓ =1 2 − 40 ǫ ≤ 2 − 9 , where we used that the maxim um of a set of nonnegativ e num b ers is at most their sum, and substituted in h ℓ = ǫ 5 2 70 ǫ ℓ , m 1 = 2 10 ǫ − 2 ≥ 2 200 , m i +1 = m i a i ≥ m i 2 ⌊ 2 − 20 m 9 / 10 i ⌋ , and t = 2 − 20 ǫ − 1 . W e thus ha v e 1 − 2 7 ν ≥ 1 / 2 ≥ ǫ r . Notice if η = ǫ r = f ( M r ), then, for 1 ≤ i ≤ r and 1 ≤ j ≤ h i , we ha v e p i,j ≥ 2 30 ǫ − 4 ǫ i ≥ 2 30 ǫ − 4 ǫ r = 2 10 η m 2 1 , where w e u s ed m 1 = 2 10 ǫ − 2 . Sin ce β = 20 m − 3 / 2 1 = 20 · 2 − 15 ǫ 3 and f (1) ≤ 2 − 100 ǫ 6 , w e ha ve δ = ǫ r = f ( M r ) ≤ f ( M 1 ) = f (1) < β / 4. By the ab o v e estimates, the conditions of Theorem 3.1 are s atisﬁed, and Corollary 4.1 stated ab o ve indeed h olds . ✷ Note that if r = t − 1 in Corollary 4.1, then |A| ≥ M t − 1 = | P t − 2 ,h t − 2 − 2 | > W , and B is a ( β , υ )- reﬁnement of P = P r,h r . As 1 − β > 1 / 2, this implies |B| ≥ 1 2 | P r,h r | > W , whic h completes the p ro of of Th eorem 1.2 in this case. W e can therefore assume r < t − 1. 4.2 Appro ximate reﬁnemen ts and mean square densit y F r om Corollary 4.1 and the follo wing lemma, we deduce at the end of this s ubsection that if P is the partition in Corollary 4.1, th en q ( P ) ≤ q ( B ) + ǫ 2 . 31 Lemma 4.1 Supp ose G is a gr aph, P is a vertex p artition, and Q is an e q u itable p artition which is a ( β , υ ) -r eﬁnement of P . Then q ( P ) ≤ q ( Q ) + 2 β + 1 2 υ . Pro of: Let Q ′ b e the common reﬁnemen t of P and Q , so q ( Q ′ ) ≥ q ( P ). Let X, Y ∈ Q b e su c h that X, Y are eac h (1 − β )-con tained in P . Let X = X 1 ∪ . . . ∪ X r b e th e partition of X consisting of parts from Q ′ with | X 1 | ≥ (1 − β ) | X | , and Y = Y 1 ∪ . . . ∪ Y s b e the p artition of Y consisting of parts from Q ′ with | Y 1 | ≥ (1 − β ) | Y | . Let p = d ( X 1 , Y 1 ) and p ′ = 1 1 − p 1 q 1 P d ( X i , Y j ) p i q j , where p i = | X i | | X | , q j = | Y j | | Y | and the sum is o v er all pairs ( i, j ) ∈ [ r ] × [ s ] except ( i, j ) = (1 , 1). That is, p ′ is th e w eigh ted a verag e edge dens ity b etw een th e pairs of p arts except ( X 1 , Y 1 ). W e ha ve r X i =1 s X j = 1 d 2 ( X i , Y j ) p i q j ≤ p 2 p 1 q 1 + X ( i,j ) 6 =(1 , 1) d ( X i , Y j ) p i q j = p 1 q 1 p 2 + p ′ (1 − p 1 q 1 ) and d ( X, Y ) = pp 1 q 1 + p ′ (1 − p 1 q 1 ) . Let ǫ = 1 − p 1 q 1 , so r X i =1 s X j = 1 d 2 ( X i , Y j ) p i q j − d 2 ( X, Y ) ≤ (1 − ǫ ) p 2 + p ′ ǫ −  p (1 − ǫ ) + p ′ ǫ  2 = ǫ  (1 − ǫ ) p 2 + p ′ − 2 pp ′ (1 − ǫ ) − p ′ 2 ǫ  ≤ ǫ  (1 − ǫ ) p 2 + p ′ − 2 pp ′ (1 − ǫ )  ≤ ǫ ≤ 2 β . The second to last in equalit y is by noting th e right hand sid e of the third to last line is linear in p ′ and m ust therefore b e maximized when p ′ = 0 or 1, and the last inequalit y follo ws from ǫ = 1 − p 1 q 1 and p 1 , q 1 ≥ 1 − β . No w for parts X , Y ∈ Q that are not b oth (1 − β )-con tained in P , again letting X = X 1 ∪ . . . ∪ X r and Y = Y 1 ∪ . . . ∪ Y s b e the partitions of X and Y consisting of parts from Q ′ , and letting q denote the edge densit y b et ween X and Y , and p i = | X i | | X | , q j = | Y j | | Y | , we ha v e r X i =1 s X j = 1 d 2 ( X i , Y j ) p i q j − d 2 ( X, Y ) ≤ q − q 2 ≤ 1 / 4 . Since Q is a ( β , υ )-reﬁnement of P , at most a 2 υ -fraction of the pairs of parts fr om Q are su c h that not b oth parts are (1 − β )-con tained in P . Putting toget her the estimates from the last t wo p aragraph s, w e therefore get q ( P ) ≤ q ( Q ′ ) ≤ q ( Q ) + 2 β + 1 4 · 2 υ . ✷ Noting th at m 1 = 2 10 ǫ − 2 , β = 20 m − 3 / 2 1 < ǫ/ 8 and υ = 5 m − 1 / 2 1 ≤ ǫ/ 4 in C orollary 4.1, we ha ve the follo wing corollary of Corolla ry 4.1 and Lemma 4.1. 32 Corollary 4.2 If P is the p artition in Cor ol lary 4.1, then q ( P ) ≤ q ( B ) + ǫ 2 . 4.3 Mean square densities of the deﬁning partitions The next lemma shows that the mean square densit y of eac h successiv e p artition increases b y a constan t factor of the edge d ensit y of eac h G i . Lemma 4.2 F or e ach i , we have q ( P i +1 ) ≥ q ( P i ) + 2 − 5 p i . Pro of: The fraction of pairs ( X, Y ) ∈ P i × P i whic h are edges of G i and not edges of G i is at least p i / 4 b y the second prop erty in Lemma 3.11. F or eac h suc h pair, the equitable partitions X = X 1 Y ∪ X 2 Y , Y = Y 1 X ∪ Y 2 X satisfy d ( X d Y , Y d X ) = 1 and d ( X d Y , Y 3 − d X ) ≤ ν ≤ 1 / 4 for d = 1 , 2. Let d 1 = d ( X 1 Y , Y 2 X ) and d 2 = d ( X 2 Y , Y 1 X ), so 2 X i =1 2 X j = 1 1 4 d 2 ( X i Y , Y j X ) − d 2 ( X, Y ) = 1 2 + d 2 1 + d 2 2 4 −  1 2 + d 1 + d 2 4  2 ≥ 1 4 − ( d 1 + d 2 ) 4 ≥ 1 8 . As we get th is d ensit y incr ement for at least a p i / 4-fractio n of the p airs ( X , Y ) ∈ P i × P i , we get a total dens ity increment of at least 1 8 p i 4 = 2 − 5 p i . ✷ W e ha v e the follo wing corolla ry , noting that p r,h r − 1 ≥ 2 10 ǫ . Corollary 4.3 F or P = P r,h r and P ′ = P r,h r − 2 , we have q ( P ) = q ( P r,h r ) ≥ q ( P r,h r − 1 ) + 2 − 5 p r,h r − 1 ≥ q ( P r,h r − 1 ) + 2 ǫ ≥ q ( P ′ ) + 2 ǫ. 4.4 Quasirandomn ess and mean squa re densit y The goal of this sub section is to show that if A is a vertex partitio n of G with |A| ≤ | P i | , then q ( A ) is at most q ( P i ) + p i plus a small err or term. T o accomplish this, we show that the graphs used to deﬁne G are quasirandom with small error. The stu dy of qu asirandom graphs b egan with the pap ers by T homason [42] and Chung, Graham, and Wilson [14]. They sho w ed th at a large n um b er of in teresting graph prop erties sati sﬁed b y rand om graphs are all equiv alen t. These prop erties are kn o wn as quasir andom pr op e rties , and any graph that has one of these prop erties (and h en ce all of these pr op erties) is known as a quasir andom gr aph . This dev elopmen t was heavi ly inﬂu enced by and closely r elated to S zemer´ edi’s r egularit y lemma. F u rthermore, all kno w n pro ofs of Szemer ´ edi’s theorem on long arithmetic progressions in dense su b sets of the in tegers use some notion of quasirandomness. F or graphs on n vertic es with edge density p b ound ed a wa y fr om zero, the follo wing t w o prop erties are quasirandom prop erties. T he ﬁr st prop ert y states th at the num b er of 4-cycles (or, equiv alen tly , the num b er of closed wal ks of length 4) in th e graph is p 4 n 4 + o ( n 4 ). The second prop erty stat es that all pairs of vertex subsets S, T hav e edge densit y roughly p b etw een them, apart from o ( n 2 ) edges. This fact, that the num b er of 4-cycles in a 33 graph can con trol th e edge d istribution, is quite notable. F or our purp oses, we will need to sho w that the ﬁrst prop ert y implies the seco nd prop ert y , w ith reasonable error esti mates. The no w standard pro of b ound s the second largest (in absolute v alue) eigen v alue of the adjacency matrix of the graph, and then app lies the expander mixin g lemma, whic h b ounds th e edge discrepancy b et w een subsets in terms o f the subset sizes and the second largest eigenv alue. Lemma 4.3 Supp ose G = ( V , E ) is a gr aph with n vertic es and aver age de gr e e d , and the numb er of close d walks of length 4 in G is at most d 4 + αn 4 . F or al l vertex subsets S and T , | e ( S, T ) − d | S || T | n | < λ p | S || T | , wher e λ ≤ α 1 / 4 n . Pro of: Let A b e th e adjace ncy matrix of G , and λ 1 , λ 2 , . . . , λ n b e the eigen v alues of A , with | λ 1 | ≥ | λ 2 | ≥ . . . ≥ | λ n | . Let λ = | λ 2 | . It is easy to c hec k that λ 1 ≥ d . Let λ = | λ 2 | . The n umber of closed w alks of length 4 in G is equ al to the trace T r ( A 4 ) = n X i =1 λ 4 i ≥ λ 4 1 + λ 4 . As λ 1 ≥ d , and the num b er of closed w alks of length 4 is at m ost d 4 + αn 4 , we conclude λ ≤ ( αn 4 ) 1 / 4 = α 1 / 4 n. The exp an d er mixing lemma (see Section 2.4 of [28]) states that for all v ertex su bsets S, T , we ha ve | e ( S, T ) − d | S | | T | n | < λ p | S || T | . T his completes the pro of. ✷ A spannin g subgraph of graph G is a sub graph of G on the same vertex set V as G . W e let H i b e the sp anning su b graph of G where vertic es u, v ∈ V are adj acent in H i if and only if there is a n edge ( X, Y ) of G i , and j ∈ { 1 , 2 } with u ∈ X j Y , v ∈ Y j X . Note that th e edge set of G is precisely the union of th e edge sets of the H i , although this is lik ely not an edge partition. W e n ext use Lemma 4.3 to sho w that the edges of H i are u niformly d istr ibuted. T hat is, the edge densit y in H i is r oughly the same b et wee n large ve rtex s u bsets of V . Lemma 4.4 L et | V | = n . F or e ach i , the gr aph H i on vertex set V ha s the pr op erty tha t for al l vertex subsets S and T , | e H i ( S, T ) − p i 2 | S || T || < 2 m − 1 / 80 i p i n p | S || T | . Pro of: Note that eac h edge ( X, Y ) of G i giv es r ise to t w o complete bipartite graph s, b et ween X j Y and Y j X with j ∈ { 1 , 2 } , in H i . In particular, eac h suc h edge of G i con trib u tes n 2 m i degree in graph H i to eac h vertex in X and in Y . W e ﬁrst giv e a low er b oun d on the a verage degree d in H i . F rom the ﬁr st prop erty in Lemma 3.11, ev ery ve rtex in G i has degree diﬀering from p i m i b y at most m 3 / 4 i . Hence, ev ery v ertex in H i has degree diﬀerin g from p i m i n 2 m i = p i n/ 2 b y at most m 3 / 4 i · n 2 m i = 1 2 m − 1 / 4 i n . Thus, the av erage degree d of H i satisﬁes | d − 1 2 p i n | ≤ 1 2 m − 1 / 4 i n . 34 W e n ext giv e an u pp er b ound on the num b er W 4 of labeled closed w alks of length four in H i . By coun ting ov er the ﬁrs t and third v ertex of the closed walk, w e h a ve W 4 = P u,v | N H i ( u, v ) | 2 , th at is, W 4 is the sum of th e squ ares of th e co degrees o v er all lab eled pairs of vertices of H i . By the ﬁrst part of Lemm a 3.11, if X , Y are distinct parts of partition P i , then the co d egree of X and Y in G i is at most p 2 i m i + m 3 / 4 i . Hence, from Corollary 2.1, if u and v are in diﬀerent parts in the partition P i , then | N H i ( u, v ) | ≤ ( 1 4 + a − 1 / 4 i )( p 2 i m i + m 3 / 4 i ) n m i = ( 1 4 + a − 1 / 4 i )( p 2 i + m − 1 / 4 i ) n. F or eac h pair of v ertices u, v in the same p art of P i , we ha v e u and v ha v e the same n eigh b orho o d in H i and in this case we use the trivial estimate | N H i ( u, v ) | ≤ n . In to tal, w e get W 4 = X u,v | N H i ( u, v ) | 2 ≤ m i ( m i − 1)  n m i  2 ·  ( 1 4 + a − 1 / 4 i )( p 2 i + m − 1 / 4 i ) n  2 + m i  n m i  2 · n 2 ≤  1 + 5 m − 1 / 20 i  p 4 i n 4 / 16 , where we used p i ≥ m − 1 / 10 i , a i = 2 ⌊ ρm 9 / 10 i ⌋ with ρ = 2 − 20 and m i ≥ m 1 ≥ 2 200 . Let α = n − 4  W 4 − d 4  ≤ n − 4  W 4 − (1 − 4 m − 1 / 4 i p − 1 i ) p 4 i n 4 / 16  ≤  5 m − 1 / 20 i + 4 m − 1 / 4 i p − 1 i  p 4 i / 16 ≤ m − 1 / 20 i p 4 i . By the c hoice of α , we ha v e W 4 = d 4 + αn 4 . F rom Lemma 4.3, we h a ve | e ( S, T ) − d | S || T | n | < α 1 / 4 n p | S || T | . Substituting in that th e av erage d egree d diﬀers from p i n/ 2 by at most m − 1 / 4 i n/ 2, the b ounds α 1 / 4 ≤ m − 1 / 80 i p i , m − 1 / 4 i / 2 ≤ m − 1 / 80 i p i , and | S || T | ≤ n p | S || T | , and u s ing the tr iangle inequalit y , w e h a ve the desired estimate holds on the n u m b er e H i ( S, T ) of edges in H i b et w een S and T . ✷ W e next p r o ve the follo win g lemma wh ich estimates the edge densit y of G b etw een certai n v ertex subsets. Lemma 4.5 L et X, Y ∈ P i b e distinct with ( X , Y ) not an e dge of G i . If also ( X , Y ) i s not an e dge of G i and A ⊂ X , B ⊂ Y , or if ( X , Y ) is an e dge of G i and ther e is j ∈ { 1 , 2 } such that A ⊂ X j Y and B ⊂ Y 3 − j X , then      d G ( A, B ) − 1 − Y h>i  1 − p h 2  !      < 6 m − 1 / 80 i +1 p i +1 n p | A || B | , wher e n = | V | is the numb er of vertic es of G . 35 Pro of: F or i ′ ≥ i , let d i ′ denote the density b et ween A and B of the pairs w hic h are edges of at least one H ℓ with ℓ ≤ i ′ . In particular, by th e c hoice of A and B , no edge s of H h for h ≤ i go b et w een A and B , and hence d i = 0. F urtherm ore, w e ha ve d i +1 = d H i +1 ( A, B ). By Lemma 4.4, the n umber of edges b et wee n A and B in H i +1 satisﬁes    e H i +1 ( A, B ) − p i +1 2 | A || B |    ≤ 2 m − 1 / 80 i +1 p i +1 n p | A || B | . (22) Let t i = 1 and for i ′ > i , let t i ′ = Q i +1 ≤ h ≤ i ′  1 − p h 2  . W e prov e b y induction on i ′ that for eac h i ′ ≥ i + 1, we h a ve | d i ′ − (1 − t i ′ ) | < q i ′ Y h = i +1 (1 + p h ) , (23) where q = 2 m − 1 / 80 i +1 p i +1 n p | A || B | . In the base case i ′ = i + 1, we h a ve the d esired estimate (23) from dividin g (22) out b y | A || B | . So supp ose we ha ve esta blished (23) for i ′ , and w e next pro v e it for i ′ + 1, completing th e pro of of (23) b y induction. Let X ′ , Y ′ ∈ P i ′ with X ′ ⊂ X , Y ′ ⊂ Y , and ( X ′ , Y ′ ) not an edge of G i ′ . If ( X ′ , Y ′ ) is n ot an edge of G i ′ , letting A ′ = X ′ ∩ A and B ′ = Y ′ ∩ B , or if ( X ′ , Y ′ ) is an edge of G i ′ , and letting j ∈ { 1 , 2 } and A ′ = X ′ j Y ∩ A and B ′ = Y ′ 3 − j X ∩ B , w e h a ve    e H i ′ +1 ( A ′ , B ′ ) − p i ′ +1 2 | A ′ || B ′ |    ≤ 2 m − 1 / 80 i ′ +1 p i ′ +1 n p | A ′ || B ′ | . Eac h suc h pair X ′ , Y ′ with ( X ′ , Y ′ ) n ot an edge of G i ′ giv es r ise to a pair ( A ′ , B ′ ), and eac h suc h pair with ( X ′ , Y ′ ) an edge of G i ′ giv es rise to tw o pairs ( A ′ , B ′ ) of this form, one for eac h j ∈ { 1 , 2 } . Th e n umber of pairs ( X ′ , Y ′ ) is  1 − d G i ′ ( X, Y )  ( m i ′ /m i ) 2 . Th e to tal num b er ∆ of suc h pairs ( A ′ , B ′ ) is therefore ∆ =  1 − d G i ′ ( X, Y ) + d G i ′ ( X, Y )  ( m i ′ /m i ) 2 . On the other h and, the sum of | A ′ || B ′ | o ver all suc h p airs is (1 − d i ′ ) | A || B | . Hence, the av erage v alue of | A ′ || B ′ | o ver a ll s u c h pairs ( A ′ , B ′ ) is (1 − d i ′ ) | A || B | / ∆ . By the triangle inequalit y , su mming o ver all such pairs A ′ , B ′ , w e ha ve the num b er of edges E o f H i ′ +1 b et w een A and B whic h are not edges of an y H ℓ with ℓ ≤ i ′ satisﬁes    E − p i ′ +1 2 (1 − d i ′ ) | A || B |    ≤ X A ′ ,B ′ 2 m − 1 / 80 i ′ +1 p i ′ +1 n p | A ′ || B ′ | ≤ 2 m − 1 / 80 i ′ +1 p i ′ +1 n ((1 − d i ′ ) | A || B | ) 1 / 2 ∆ 1 / 2 ≤ 4 m − 1 / 80 i ′ +1 p i ′ +1 n ( | A || B | ) 1 / 2 m i ′ /m i , where we used J ensen’s inequalit y for th e conca ve function f ( y ) = y 1 / 2 . 36 Hence, | d i ′ +1 − (1 − t i ′ +1 ) | =     d i ′ ( A, B ) + E | A || B | − (1 − t i ′ +1 )     ≤ | d i ′ − (1 − t i ′ ) | +     E | A || B | − ( t i ′ − t i ′ +1 )     = | d i ′ − (1 − t i ′ ) | +     E | A || B | − p i ′ +1 2 t i ′     ≤ (1 + p i ′ +1 2 ) | d i ′ − (1 − t i ′ ) | +     E | A || B | − p i ′ +1 2 (1 − d i ′ )     ≤ (1 + p i ′ +1 2 ) | d i ′ − (1 − t i ′ ) | + 4 m − 1 / 80 i ′ +1 p i ′ +1 n ( | A || B | ) − 1 / 2 m i ′ /m i ≤ q  1 + p i ′ +1 2  i ′ Y h = i +1 (1 + p h ) + 4 m − 1 / 80 i ′ +1 p i ′ +1 n ( | A || B | ) − 1 / 2 m i ′ /m i ≤ q i ′ +1 Y h = i +1 (1 + p h ) , whic h completes the indu ction pro of of (23). As P p h ≤ 1, w e hav e Q (1 + p h ) ≤ e . F rom (23) with i ′ = s − 1, w e get      d G ( A, B ) − 1 − Y h>i (1 − p h ) !      = | d s − 1 − (1 − t s − 1 ) | < q s − 1 Y h = i +1 (1 + p h ) < 6 m − 1 / 80 i +1 p i +1 n p | A || B | , whic h completes the pr o of. ✷ The follo w ing lemma is the main result in this su bsection, showing that q ( A ) − q ( P ′ ) is small, where the mean square densities are with resp ect to the graph G . Lemma 4.6 F or P ′ = P r,h r − 2 , we have q ( A ) ≤ q ( P ′ ) + ǫ 2 . Pro of: Consider th e p artition A ′ whic h is the common reﬁn emen t of P ′ and A . The num b er of p arts of A ′ is at most | P ′ ||A| ≤ | P ′ | 2 , and eac h part of P ′ is reﬁned into at most |A| ≤ | P ′ | p arts of A ′ . Let i b e suc h th at P i = P r,h r − 2 = P ′ . As A ′ is a reﬁnemen t of P ′ , in H j for eac h j < i b et ween eac h pair of p arts of A ′ the edge d ensit y is 0 o r 1. Noting th at A ′ is a reﬁnemen t of A , we ha ve q ( A ) − q ( P i ) ≤ q ( A ′ ) − q ( P i ) = X X,Y ∈ P i m − 2 i X A,B ⊂A ′ ,A ⊂ X, B ⊂ Y | A || B | | X || Y |  d 2 ( A, B ) − d 2 ( X, Y )  . (24) Note that the summand in the ab ov e su m if ( X , Y ) is an edge of G i is 0 as in this case d ( A, B ) = d ( X, Y ) = 1. W e hav e d 2 ( A, B ) − d 2 ( X, Y ) ≤ 1 for ( X, Y ) an edge of G i , and the fr action of p airs ( X, Y ) which are edges of G i is at most p i + m − 1 / 4 i . F or a pair X , Y ∈ P i with ( X, Y ) not an edge of G i or G i , A, B ⊂ A ′ with A ⊂ X and B ⊂ Y , we ha ve by Lemma 4.5 and the triangle inequalit y that | d ( A, B ) − d ( X, Y ) | ≤ 2 · 6 m − 1 / 80 i +1 p i +1 n p | A || B | . (25) 37 Summing o v er all parts A, B of A ′ with A ⊂ X and B ⊂ Y , w e ha ve X A,B ⊂A ′ ,A ⊂ X, B ⊂ Y | A || B |  d 2 ( A, B ) − d 2 ( X, Y )  ≤ X A,B ⊂A ′ ,A ⊂ X, B ⊂ Y | A || B | 2 | d ( A, B ) − d ( X, Y ) | ≤ X A,B ⊂A ′ ,A ⊂ X, B ⊂ Y 24 m − 1 / 80 i +1 p i +1 n p | A || B | ≤ 24 m − 1 / 80 i +1 p i +1 n 2 , where the ﬁrst inequalit y follo ws from a 2 − b 2 = ( a + b )( a − b ) ≤ 2( a − b ) f or 0 ≤ a, b ≤ 1, the second inequalit y is by (25), an d the last inequality is b y using the Cauc h y-Sc h w arz inequalit y , noting that X A,B ⊂A ′ ,A ⊂ X, B ⊂ Y | A || B | = | X || Y | = ( n/m i ) 2 , and the n u m b er of pairs A, B ⊂ A ′ satisfying A ⊂ X , B ⊂ Y is at most m 2 i . Dividing out b y | X || Y | = ( n/m i ) 2 , we h a ve, X A,B ⊂A ′ ,A ⊂ X, B ⊂ Y | A || B | | X || Y |  d 2 ( A, B ) − d 2 ( X, Y )  ≤ 24 m − 1 / 80 i +1 p i +1 m 2 i . (26) F r om the estimate (26), we hav e from (24 ) that q ( A ) − q ( P ′ ) ≤ p i + m − 1 / 4 i + 24 m − 1 / 80 i +1 p i +1 m 2 i ≤ 3 p i ≤ ǫ 2 , (27) where we used p i = max( m − 1 / 10 i , 2 30 ǫ − 4 ǫ r ) , ǫ r ≤ ǫ 1 = f (1) = 2 − 100 ǫ 6 , i ≥ h 1 − 2 = ǫ 5 2 70 ǫ 1 − 2 ≥ 2 29 ǫ − 1 and hen ce m i ≥ (6 /ǫ ) 10 . T his completes the pro of. ✷ 5 Induced graph remo v al lemma The induced graph remo v al lemma states th at for an y ﬁxed graph H on h vertices and ǫ > 0, there is δ = δ ( ǫ, H ) > 0 suc h that if a graph G on n v ertices has at most δ n h induced copies of H , then we can add or delete ǫn 2 edges of G to obtain an induced H -free graph. T he main goal of this section is to pro v e T h eorem 1.3, which give s a b ound on δ − 1 whic h is a to wer in h of heigh t p olynomial in ǫ − 1 . W e in fac t pro ve the k ey co rollary of the strong regularit y lemma, Lemma 1.2, w ith a to wer-t yp e b ound. This is suﬃcien t to pro v e the desired to w er-t yp e b ou n d f or the induced grap h remo v al lemma. W e ﬁrst use the wea k regularit y lemma o f Duke, Lefmann, and R¨ odl to ﬁnd a large sub s et of a graph whic h is ǫ -r egular with itself. B y iterativ ely pulling out suc h sub sets and redistributing the set of 38 lefto ve r vertice s, w e obtain a partition of an y v ertex subset into large v ertex subsets ea c h of w hic h is ǫ -regular with itself. Then, in Su bsection 5.4, w e esta blish Lemma 1.3, the strong cylinder regularit y lemma, with a to w er-t yp e b oun d. W e show in Su bsection 5.5 that the strong cylinder regularit y lemma implies the key corollary of th e strong regularit y lemma, Lemma 1.2, w ith a to w er-t yp e b ound. Th is in turn implies Theorem 1.3. In this s ection and the next, we call a p air ( A, B ) of v ertex subsets of a graph ǫ - r e g u lar if f or all A ′ ⊂ A and B ′ ⊂ B with | A ′ | ≥ ǫ | A | and | B ′ | ≥ ǫ | B | , w e ha ve | d ( A ′ , B ′ ) − d ( A, B ) | ≤ ǫ . 5.1 The Duke- Lefmann-R¨ odl r egularity lemma Giv en a k -partite graph G = ( V , E ) with k -partition V = V 1 ∪ . . . ∪ V k , we will consider a p artition K of the cylinder V 1 × · · · × V k in to cylinders K = W 1 × · · · × W k , W i ⊂ V i for i = 1 , . . . , k , and w e let V i ( K ) = W i . Recall f rom the introd uction th at a cylinder is ǫ -regular if all the  k 2  pairs of subsets ( W i , W j ), 1 ≤ i < j ≤ k , are ǫ -regular. The partition K is ǫ -regular if all but an ǫ -fraction of the k -tuples ( v 1 , . . . , v k ) ∈ V 1 × · · · × V k are in ǫ -regular cylind ers in the partition K . The w eak regularity lemma of Duke , Lefmann, and R¨ odl [15] is no w as follo ws. Note that, lik e the F r ieze-Kannan weak regularit y lemma, it has only a single-exp onential b ound on th e num b er of p arts, whic h is m u c h b etter than the tow er-t yp e b ound on the n umb er of parts in Szemer´ edi’s regularity lemma. Duk e, Lefmann, and R¨ odl [15] used their regularit y lemma to deriv e a fast appr o ximation algorithm for the n umb er of copies of a ﬁxed graph in a large graph. Lemma 5.1 L et 0 < ǫ < 1 / 2 and β = β ( ǫ ) = ǫ k 2 ǫ − 5 . Supp ose G = ( V , E ) is a k -p artite gr aph with k -p artition V = V 1 ∪ . . . ∪ V k . Then ther e exists an ǫ -r e gular p artition K of V 1 × · · · × V k into at most β − 1 p arts such that, for e ach K ∈ K and 1 ≤ i ≤ k , we have | V i ( K ) | ≥ β | V i | . 5.2 Finding an ǫ -regular subset F or a graph G = ( V , E ), a v ertex sub s et U ⊂ V is ǫ -r e gular if the pair ( U, U ) is ǫ -regular. The follo wing lemma demons tr ates that any graph co nt ains a large v ertex subset which is ǫ -regular. Lemma 5.2 F or e ach 0 < ǫ < 1 / 2 , let δ = δ ( ǫ ) = 2 − ǫ − (10 /ǫ ) 4 . Every gr aph G = ( V , E ) c ontains an ǫ -r e gular vertex subset U with | U | ≥ δ | V | . Lemma 5.1 implies that eac h k -partite graph G = ( V , E ) with k -p artition V = V 1 ∪ . . . ∪ V k has a cylinder K wh ic h is ǫ -r egular in whic h eac h part h as size | V i ( K ) | ≥ ǫ k 2 ǫ − 5 | V i | . The pro of can b e easily mo diﬁed to s h o w that if eac h p art of G has the same size, then eac h part of the ǫ -r egular cylinder K has equal size, whic h is at lea st ǫ k 2 ǫ − 5 | V i | . This implies th at for an y graph G = ( V , E ), if G h as at least k v ertices, b y co nsidering an y k ve rtex disjoin t subs ets of equal size ⌊| G | / k ⌋ ≥ | G | / (2 k ), and then applyin g this result, we get the follo wing lemma. Lemma 5.3 F or e ach 0 < ǫ < 1 / 2 , any gr aph G = ( V , E ) on at le ast k vertic es has an ǫ -r e gular k -cylinder with p arts of e qual size, which is at le ast 1 2 k ǫ k 2 ǫ − 5 | V | . 39 The t -c olor R amsey numb er r t ( s ) is the minim u m k suc h that eve ry t -coloring of the edges of the complete graph K k on k v ertices con tains a m on o c hromatic clique of order s . A simple pigeonhole argumen t (see [23]) giv es r t ( s ) ≤ t ts for t ≥ 2. Lemma 5.4 F or inte gers s, t ≥ 2 , let k = t ts . L e t G = ( V , E ) b e a gr aph on at le ast k vertic es, and 0 < α < 1 / 2 . The gr aph G c ontains an α -r e gular s -cylinder with p arts of e qual size at le ast N = 1 2 k α k 2 α − 5 | V | such that the density b etwe en e ach p air of p arts diﬀers by at most 1 /t . Pro of: No te that k = t ts ≥ r t ( s ). By Lemma 5.3, G con tains an α -regular k -cylinder U 1 × · · · × U k with parts of equal size at least N . P artition the unit int erv al [0 , 1] = I 1 ∪ . . . ∪ I t in to t in terv als of length 1 /t . Consider the edge-coloring of th e complete graph on k v ertices 1 , . . . , k for wh ic h the color of edge ( i, j ) is the n umber a for whic h the density d ( U i , U j ) ∈ I a . Since k ≥ r t ( s ), th er e is a monochromatic clique of order s in this t -colo ring, and the co rresp onding parts form the d esired s -cylinder. ✷ Lemma 5.5 Supp ose α ≤ 1 / 9 and U 1 × · · · × U s is an α -r e gular cylinder in a gr aph G = ( V , E ) with s ≥ 2 α − 1 p arts U i of e qual size and the densities b etwe en the p airs of distinct p arts lie in an interval of length at most α . Then the set U = U 1 ∪ . . . ∪ U s is ǫ -r e gular with itself, wher e ǫ = 3 α 1 / 2 . Pro of: Let A, B ⊂ U with | A | , | B | ≥ ǫ | U | and, for 1 ≤ i ≤ s , let A i = A ∩ U i and B i = B ∩ U i . Supp ose d ( U i , U j ) ∈ [ γ , γ + α ] for 1 ≤ i < j ≤ s . Let A 1 b e the union of all A i for whic h | A i | ≥ α | U i | and A 2 = A \ A 1 . Similarly , let B 1 b e th e union of all B i for whic h | B i | ≥ α | U i | and B 2 = B \ B 1 . W e ha ve | A 2 | < α | U | ≤ αǫ − 1 | A | and | B 2 | < α | U | ≤ αǫ − 1 | B | . Let I 1 denote the set of all pairs ( i, i ) with i ∈ [ s ], I 2 the set of all pairs ( i, j ) ∈ [ s ] × [ s ] with i 6 = j , A i ⊂ A 1 , and B j ⊂ B 1 , an d I 3 = [ s ] × [ s ] \ ( I 1 ∪ I 2 ). Let D ( A i , B j ) = | d ( A i , B j ) − γ | | A i || B j | | A || B | . W e hav e X ( i,i ) ∈ I 1 D ( A i , B i ) ≤ X ( i,i ) ∈ I 1 | A i || B i | | A || B | ≤ max i | B i | | B | ≤ max i | U i | | B | ≤ 1 sǫ . If ( i, j ) ∈ I 2 , using the triangle inequalit y and α -regularit y , | d ( A i , B j ) − γ | ≤ | d ( A i , B j ) − d ( U i , U j ) | + | d ( U i , U j ) − γ | ≤ α + α = 2 α. Hence, X ( i,j ) ∈ I 2 D ( A i , B j ) ≤ X ( i,j ) ∈ I 2 2 α | A i || B j | | A || B | ≤ 2 α. Finally , X ( i,j ) ∈ I 3 D ( A i , B j ) ≤ X ( i,j ) ∈ I 3 | A i || B j | | A || B | ≤ 1 −  1 − | A 2 | | A |   1 − | B 2 | | B |  < 1 − (1 − αǫ − 1 ) 2 ≤ 2 αǫ − 1 . 40 W e ha v e by the triangle inequalit y | d ( A, B ) − γ | ≤ X 1 ≤ i,j ≤ s D ( A i , B j ) = X ( i,j ) ∈ I 1 D ( A i , B j ) + X ( i,j ) ∈ I 2 D ( A i , B j ) + X ( i,j ) ∈ I 3 D ( A i , B j ) ≤ 1 sǫ + 2 α + 2 αǫ − 1 ≤ ǫ 2 . By the triangle inequalit y , for a ny A, B , X , Y ⊂ U eac h of cardinalit y at least ǫ | U | , we ha v e | d ( A, B ) − d ( X, Y ) | ≤ | d ( A, B ) − γ | + | γ − d ( X, Y ) | ≤ ǫ 2 + ǫ 2 = ǫ. In p articular, this holds f or X = Y = U , and hence U is ǫ -regular. ✷ By applying Lemma 5.4 with α = ( ǫ/ 3) 2 and s = t = ⌈ 2 α − 1 ⌉ , and then app lyin g Lemma 5.5, w e obtain Lemma 5.2. Note that the pro of assumes th at the num b er of v ertices of th e graph is suﬃcien tly large, at least k = t ts , b ut w e can mak e this assu mption as otherw ise w e can trivially pic k U to consist of a single verte x, whic h is ǫ -regular. The next lemma shows that if w e h av e an ǫ -regular pair, and add a small f r action of v ertices to one part, th en the pair is still r egular, bu t with a sligh tly w orse regularit y . Lemma 5.6 Supp ose A and B ar e ve rtex subsets of a gr aph G which form an ǫ -r e gular p air, and C is a vertex subset disjoint fr om B of size | C | ≤ β | B | . Th en the p air ( A, B ∪ C ) is α - r e gular with α = ǫ + √ β + β . Pro of: Let A ′ ⊂ A and B ′ ∪ C ′ ⊂ B ∪ C with B ′ ⊂ B , C ′ ⊂ C , | A ′ | ≥ α | A | and | B ′ ∪ C ′ | ≥ α | B ∪ C | . Note that | A ′ | ≥ α | A | ≥ ǫ | A | and | B ′ | = | B ′ ∪ C ′ | − | C ′ | ≥ α | B ∪ C | − | C | ≥ ( α − β ) | B | ≥ ǫ | B | . Since the p air ( A, B ) is ǫ -regular, we h a ve   d ( A ′ , B ′ ) − d ( A, B )   ≤ ǫ. Also, | C ′ | ≤ | C | ≤ β | B | ≤ β | B ∪ C | ≤ β α − 1 | B ′ ∪ C ′ | . Therefore, | d ( A ′ , B ′ ∪ C ′ ) − d ( A ′ , B ′ ) | =     d ( A ′ , B ′ ) | B ′ | | B ′ ∪ C ′ | + d ( A ′ , C ′ ) | C ′ | | B ′ ∪ C ′ | − d ( A ′ , B ′ )     =   d ( A ′ , C ′ ) − d ( A ′ , B ′ )   | C ′ | | B ′ ∪ C ′ | ≤ | C ′ | | B ′ ∪ C ′ | ≤ β α − 1 . W e similarly ha v e | d ( A, B ∪ C ) − d ( A, B ) | = | d ( A, C ) − d ( A, B ) | | C | | B ∪ C | ≤ β . Hence, by the tria ngle in equalit y , w e ha v e | d ( A ′ , B ′ ∪ C ′ ) − d ( A, B ∪ C ) | is at most   d ( A ′ , B ′ ∪ C ′ ) − d ( A ′ , B ′ )   +   d ( A ′ , B ′ ) − d ( A, B )   + | d ( A, B ) − d ( A, B ∪ C ) | ≤ β α − 1 + ǫ + β ≤ α. 41 Hence, th e pair ( A, B ∪ C ) is α -regular. ✷ By rep eatedly pulling out 3 ǫ/ 4-regular sets using Lemma 5.2 unt il there are at most ǫ 2 100 | V | remaining v ertices, distribu ting the remaining ve rtices among the parts, and u s ing Lemma 5.6 t wice in eac h part, w e arrive at the follo w ing lemma. It shows ho w to partition a g raph into large p arts, eac h part b eing ǫ -regular w ith itself. Lemma 5.7 F or e ach 0 < ǫ < 1 / 2 , let δ = δ ( ǫ ) = 2 − ǫ − (20 /ǫ ) 4 . Every gr aph G = ( V , E ) has a vertex p artition V = V 1 ∪ . . . ∪ V k such that for e ach i , 1 ≤ i ≤ k , we have | V i | ≥ δ | V | and V i is an ǫ -r e gular set. 5.3 T o ols In this su bsection, w e pro ve tw o simple le mmas concerning mean squ are dens it y whic h will b e us eful in establishing and using th e strong cyl inder regularit y lemma. The ﬁr st lemma, wh ic h is r ather standard, sho w s that for any v ertex partition P , there is a vertex equipartition P ′ with a similar num b er of parts to P and mean square dens it y not m u c h s m aller than the mean squ are d en sit y of P . It is useful in densit y in cr ement argum en ts where at eac h stag e on e w ould like to work with a n equ ip artition. Lemma 5.8 L et G = ( V , E ) b e a gr aph, and P : V = V 1 ∪ . . . ∪ V k b e a vertex p artition into k p arts. Ther e is an e qu itable p artition P ′ of V into t p arts such that q ( P ′ ) ≥ q ( P ) − 2 k t . Pro of: F or an equip artition of V into t parts, we h av e a certain num b er of parts of ord er ⌊| V | /t ⌋ and the remaining parts are of ord er ⌈| V | /t ⌉ . F or eac h part V i ∈ P , partition it into parts of order ⌊| V | /t ⌋ or ⌈| V | /t ⌉ so that there are not to o m any parts of either order to allo w an equipartition of the whole set, with p ossib ly one r emaining set of cardinalit y less than | V | /t . L et Q b e this r eﬁnemen t of P . F rom the Cauch y-Sc hw arz inequalit y , it follo ws that q ( Q ) ≥ q ( P ). Let U b e the v ertices in the remaining p arts of Q , so | U | < k | V | /t . Arbitrarily chop th e v ertices of U in to p arts of the desired ord ers so as to obta in an equip artition P ′ . W e ha v e q ( P ′ ) ≥ X X,Y ∈ Q,X,Y ⊂ V \ U d 2 ( X, Y ) | X || Y | | V | 2 ≥ q ( Q ) − 1 −  1 − | U | | V |  2 ! ≥ q ( Q ) − 2 k t ≥ q ( P ) − 2 k t . ✷ The next lemma is helpful in deducing the induced graph remo v al lemma f rom the strong cylind er regularit y lemma. Let G = ( V , E ) and P : V = V 1 ∪ . . . ∪ V k b e an equip artition, and K b e a partition of the cylinder V 1 × · · · × V k in to cylinders. F or K = W 1 × · · · × W k ∈ K , deﬁ ne the den sit y d ( K ) = | W 1 |×···×| W k | | V 1 |×···×| V k | . T h e cylinder K is ǫ -close to P if | d ( W i , W j ) − d ( V i , V j ) | ≤ ǫ for all but at most ǫk 2 pairs 1 ≤ i 6 = j ≤ k . if cylinder K is not ǫ -close to P , then X 1 ≤ i 6 = j ≤ k | d ( W i , W j ) − d ( V i , V j ) | > ǫ 2 k 2 . 42 The cylinder partition K is ǫ -close to P if P d ( K ) ≤ ǫ , where the sum is ov er all K ∈ K that are not ǫ -close to P . Note th at if K is not ǫ -close, then X K ∈K X 1 ≤ i 6 = j ≤ k | d ( W i , W j ) − d ( V i , V j ) | d ( K ) > ǫ 3 k 2 . Recall that Q ( K ) is the common reﬁnement of all the parts V i ( K ) with i ∈ [ k ] and K ∈ K . Lemma 5.9 L et G = ( V , E ) and P : V = V 1 ∪ . . . ∪ V k b e an e qu ip artition with no empty p arts, i . e., | V | ≥ k . L et K b e a p artition of the cylinder V 1 × · · · × V k into cylinders. If Q = Q ( K ) satisﬁes q ( Q ) ≤ q ( P ) + ǫ , then K is 2 1 / 3 ǫ 1 / 6 -close to P . Pro of: Let Q i denote the partitio n of V i whic h is the restriction of p artition Q to V i . Since P is an equipartition and | V | ≥ k , then all parts ha v e ord er at least j | V | k k ≥ | V | 2 k . T herefore, ǫ ≥ q ( Q ) − q ( P ) = X 1 ≤ i,j ≤ k ( q ( Q i , Q j ) − q ( V i , V j )) | V i || V j | | V | 2 ≥ 1 4 k 2 X 1 ≤ i 6 = j ≤ k ( q ( Q i , Q j ) − q ( V i , V j )) , (28) where q ( Q i , Q j ) = P A ∈ Q i ,B ∈ Q j d 2 ( A, B ) p A p B with p A = | A | | V i | and p B = | B | | V j | , an d q ( V i , V j ) = d 2 ( V i , V j ). Fix for no w 1 ≤ i 6 = j ≤ k . F or K = W 1 × · · · × W k ∈ K , we ha v e d ( W i , W j ) = X d ( A, B ) | A | | W i | | B | | W j | , and hen ce, by the triangle inequalit y , | d ( W i , W j ) − d ( V i , V j ) | ≤ X | d ( A, B ) − d ( V i , V j ) | | A | | W i | | B | | W j | , where the sums are o ver all A ∈ Q i with A ⊂ W i and B ∈ Q j with B ⊂ W j . Sum ming o v er all K ∈ K , w e hav e, X K = W 1 ×··· W k ∈K | d ( W i , W j ) − d ( V i , V j ) | d ( K ) ≤ X K = W 1 ×··· W k ∈K X | d ( A, B ) − d ( V i , V j ) | | A | | W i | | B | | W j | d ( K ) = X A ∈ Q i ,B ∈ Q j | d ( A, B ) − d ( V i , V j ) | p A p B ≤   X A ∈ Q i ,B ∈ Q j ( d ( A, B ) − d ( V i , V j )) 2 p A p B   1 / 2 = ( q ( Q i , Q j ) − q ( V i , V j )) 1 / 2 . where the ﬁrst equalit y follo ws by sw app ing the order of summation and the last in equ alit y is th e Cauc hy-Sc hw arz inequalit y . 43 Summing o v er all 1 ≤ i 6 = j ≤ k and changing the order of summation, X K = W 1 ×··· W k ∈K X 1 ≤ i 6 = j ≤ k | d ( W i , W j ) − d ( V i , V j ) | d ( K ) ≤ X 1 ≤ i 6 = j ≤ k ( q ( Q i , Q j ) − q ( V i , V j )) 1 / 2 ≤   k 2 X 1 ≤ i 6 = j ≤ k q ( Q i , Q j ) − q ( V i , V j )   1 / 2 ≤ √ k 2 · 4 k 2 ǫ = 2 ǫ 1 / 2 k 2 , where the second in equalit y is the Cauc h y-Sc hw arz in equ alit y and the last in equalit y uses th e estimate (28). By the remark b efore the lemma, w e ge t that K is  2 ǫ 1 / 2  1 / 3 = 2 1 / 3 ǫ 1 / 6 -close to P . ✷ 5.4 The str ong cylinder regularit y lemma Using the lemmas established in the previous sub sections, in th is sub section w e pr ov e Lemma 1.3, the strong cylind er regularit y lemma, with a to w er-t yp e b ound. Recall that a k -cylind er W 1 × · · · × W k is strongly ǫ -regular if all pairs ( W i , W j ) with 1 ≤ i, j ≤ k are ǫ -regular. A p artition K of V 1 × · · · × V k in to cylinders is strongly ǫ -regular if all but ǫ | V 1 | × · · · × | V k | v ertices ( v 1 , . . . , v k ) ∈ V 1 × · · · × V k are cont ained in strongly ǫ -regular cylinders K ∈ K . W e recall the statemen t of the strong cylinder regularit y lemma. Lemma 5.10 F or 0 < ǫ < 1 / 3 , p ositive inte ger s , and de cr e asing function f : N → (0 , ǫ ] , ther e is S = S ( ǫ, s, f ) such that the fol lowing holds. F or every gr aph G , ther e is an inte ger s ≤ k ≤ S , an e qui table p artition P : V = V 1 ∪ . . . ∪ V k and a str ongly f ( k ) -r e gular p artition K of the c ylinder V 1 × · · · × V k into cylinders satisfying that the p artition Q = Q ( K ) of V has at most S p arts and q ( Q ) ≤ q ( P ) + ǫ . F urthermor e, ther e is an absolute c onstant c su c h that letting s 1 = s and s i +1 = t 4 (( s i /f ( s i )) c ) , we may take S = s ℓ with ℓ = 2 ǫ − 1 + 1 . Pro of: W e ma y assume | V | ≥ S , as otherwise we can let P and Q b e the trivial partitions into singletons, and it is easy to see the lemma holds. W e will deﬁne a sequ ence of partitions P 1 , P 2 , . . . of equitable partitio ns, with P j + 1 a reﬁn emen t of P j and q ( P j + 1 ) > q ( P j ) + ǫ/ 2. Let P 1 b e an arbitrary equitable partition of V consisting of s 1 = s parts. Supp ose w e ha ve already found an equitable partition P j : V = V 1 ∪ . . . ∪ V k with k ≤ s j . Let β ( x, ℓ ) = x ℓ 2 x − 5 as in L emm a 5.1 an d δ ( x ) = 2 − x − (20 /x ) 4 as in L emm a 5.7. W e apply Lemma 5.7 to eac h part V i of the partitio n P j to get a partition of eac h part V i = V i 1 ∪ . . . ∪ V ih i of P i in to p arts eac h of cardinalit y at least δ | V i | , where δ = δ ( γ ) and γ = f ( k ) · β with β = β ( f ( k ) , k ), suc h that eac h part V ih is γ -regular. Note th at δ − 1 is at most triple-exp onenti al in a p olynomial in k /f ( k ). F or eac h k -tuple ℓ = ( ℓ 1 , . . . , ℓ k ) ∈ [ h 1 ] × · · · × [ h k ], b y Lemm a 5.1 there is an f ( k )-regular partition K ℓ of the cylinder V 1 ℓ 1 × · · · × V k ℓ k in to at most β − 1 cylinders such that, for eac h K ∈ K ℓ , | V iℓ i ( K ) | ≥ β | V iℓ i | . The un ion of the K ℓ forms a partition K of V 1 × · · · × V k whic h is strongly f ( k )-regular. 44 Recall that Q = Q ( K ) is the p artition of V whic h is the common reﬁnement of all p arts V i ( K ) with i ∈ [ k ] and K ∈ K . The num b er of parts of K is at most δ − k β − 1 , an d hence the num b er of parts of Q is at most k 2 1 / ( δ k β ) . Th us, the num b er of parts of Q is at most quadrup le-exp on ential in a p olynomial in k /f ( k ). Let P j + 1 b e an equitable partition in to 4 ǫ − 1 | Q | p arts with q ( P j + 1 ) ≥ q ( Q ) − ǫ 2 , w h ic h exists b y Lemma 5.8. Hence, there is an absolute constan t c suc h that | P j + 1 | ≤ t 4 (( k /f ( k )) c ) ≤ s j + 1 . If q ( Q ) ≤ q ( P j ) + ǫ , then we ma y tak e P = P j and Q = Q ( K ), and these partitions satisfy the desired prop erties. O therwise, q ( P j + 1 ) ≥ q ( Q ) − ǫ 2 > q ( P j ) + ǫ 2 , and w e con tinue the sequence of partitions. Since q ( P 1 ) ≥ 0, and the mean square dens ity go es up b y more than ǫ/ 2 at eac h step and is alwa ys at most 1, this pro cess must stop w ithin 2 /ǫ steps, and w e obtain the d esired partitions. ✷ 5.5 A to wer- ty p e bound for the key corollary In th e previous su bsection, w e established the strong cylinder regularit y lemma with a to w er-t yp e b ound . W e next u se this result to dedu ce a to wer-t yp e b ound for Lemma 1.2, whic h is th e k ey corollary of the strong regularity lemma, and easily imp lies the induced graph r emov al lemma as sho wn b elo w. W e recall the statemen t of Lemma 1.2 b elo w. Lemma 5.11 F or e ach 0 < ǫ < 1 / 3 and de cr e asing function f : N → (0 , ǫ ] ther e is δ ′ = δ ′ ( ǫ, f ) such that every gr aph G = ( V , E ) with | V | ≥ δ ′− 1 has an e quitable p artition V = V 1 ∪ . . . ∪ V k and v ertex subsets W i ⊂ V i such that | W i | ≥ δ ′ | V | , e ach p air ( W i , W j ) with 1 ≤ i ≤ j ≤ k is f ( k ) -r e gular, and al l but at most ǫk 2 p airs 1 ≤ i ≤ j ≤ k satisfy | d ( V i , V j ) − d ( W i , W j ) | ≤ ǫ . F urthermor e, we may take δ ′ = 1 8 S 2 , wher e S = ( ǫ 6 4 , s, f ) is deﬁne d as in L emma 5.10 and s = 2 ǫ − 1 . Pro of: Let α = ǫ 6 4 , s = 2 ǫ − 1 , and δ ′ = 1 8 S 2 , wh ere S = S ( α, s , f ) is as in Lemma 5.10. W e apply Lemma 5.10 with α in p lace of ǫ . W e get an equipartition P : V = V 1 ∪ . . . ∪ V k with s ≤ k ≤ S and a strongly f ( k )-regular partition K of V 1 × · · · × V k in to cylinders such that the reﬁnement Q = Q ( K ) of P has at m ost S = S ( α, s, f ) parts and sati sﬁes q ( Q ) ≤ q ( P ) + α . Since | V | ≥ δ ′− 1 = 8 S 2 , and P is an equipartition in to k ≤ S parts, the cardinalit y of eac h part V i ∈ P satisﬁes | V i | ≥ | V | 2 S . By Lemma 5.9, as 2 1 / 3 α 1 / 6 = ǫ , the cylinder partition K is ǫ -close to P . Hence, at most an ǫ -fr action of the k -tup les ( v 1 , . . . , v k ) ∈ V 1 × · · · × V k b elong to parts K = W 1 × · · · × W k of K that are not ǫ -close to P . Since Q ( K ) has at most S parts, the fraction of k -tup les ( v 1 , . . . , v k ) ∈ V 1 × · · · × V k that b elong to parts K = W 1 × · · · × W k of K with | W i | < 1 4 S | V i | f or at least one i ∈ [ k ] is at most 1 4 S · S = 1 4 . Therefore, at least a fraction 1 − f ( k ) − ǫ − 1 4 > 0 of the k -tuples ( v 1 , . . . , v k ) ∈ V 1 × · · · × V k b elong to parts K = W 1 × · · · × W k of K satisfying K is strongly f ( k )-regular, | W i | ≥ 1 4 S | V i | ≥ δ ′ | V | for i ∈ [ k ], and K is ǫ -close to P . Since a p ositive fraction of the k -tup les b elong to suc h K , there is at least one suc h K . This K has the desired prop erties. Ind eed the n umb er of pairs 1 ≤ i 6 = j ≤ k for which | d ( W i , W j ) − d ( V i , V j ) | > ǫ is at most ǫk 2 and hence the n umb er of pairs 1 ≤ i ≤ j ≤ k for w hic h | d ( W i , W j ) − d ( V i , V j ) | > ǫ is at most ǫk 2 / 2 + k ≤ ǫk 2 . T his completes the pro of. ✷ 45 W e next d iscu ss ho w to obtain the in d uced graph remov al lemma from Lemma 1.2. This is a b it easier to obtain than in [5] b ecause L emm a 1.2 has the add itional pr op ert y that th e subs ets W i in the cylinder K are ǫ -regular. W e ﬁnish this section b y giving this pr o of and discussing the b oun d it giv es for the ind u ced graph remo v al lemma, wh ic h is a to wer in h of h eigh t p olynomial in ǫ − 1 . W e ﬁr st need the follo wing counting lemma, whic h is rather standard (see, e.g., Lemma 3.2 in Alon, Fisc her, Kriv elevic h, and S zegedy [5] for a minor v ariant). W e omit the p ro of. Lemma 5.12 If H is a gr aph with vertic es 1 , . . . , h , and G is a gr aph with not ne c essarily disjoint vertex subsets W 1 , . . . , W h such that ev ery p air ( W i , W j ) with 1 ≤ i < j ≤ h is γ -r e gular with γ ≤ 1 4 h η h , | W i | ≥ γ − 1 for 1 ≤ i ≤ h and, for 1 ≤ i < j ≤ k , d ( W i , W j ) > η if ( i, j ) is an e dge of H and d ( W i , W j ) < 1 − η otherwise, then G c ontains at le ast  η 4  ( h 2 ) | W 1 | × · · · × | W h | induc e d c opies of H with the c opy of vertex i in W i . W e ﬁnish th e section with a quan titativ e v ers ion of Theorem 1.3. Theorem 5.1 Ther e is an absolute c onstant c suc h that for any gr aph H on h vertic es 1 , . . . , h and 0 < ǫ < 1 / 2 ther e is δ > 0 with δ − 1 = t j ( h ) with j = O ( ǫ − 6 ) such that if a gr aph G on n vertic es has at most δ n h induc e d c opies of H , then we c an add or delete ǫ n 2 e dges of G to obtain an i nduc e d H -fr e e gr aph. Pro of: Let η = ǫ 8 . Let δ =  η 4  h 2 δ ′ h , where δ ′ = δ ′ ( η , f ) as in Lemma 5.11 and f ( k ) = 1 4 h η h . If the n umber of v ertices satisﬁes | V | < δ − 1 /h , then δ | V | h < 1 and there are no ind uced copies of H , in wh ic h case no edges of G need to b e mo diﬁed . W e can therefore assum e that | V | ≥ δ − 1 /h =  η 4  − h δ ′− 1 . As | V | ≥ δ ′− 1 , we can apply Lemma 5.11 to graph G = ( V , E ) w ith η in place of ǫ and f as ab o ve. W e obtain an equitable partition V = V 1 ∪ . . . ∪ V k and v ertex subsets W i ⊂ V i suc h that | W i | ≥ δ ′ | V | ≥  η 4  − h , the cylinder W 1 × · · · W k is strongly f ( k )-regular, and all but at most η k 2 pairs 1 ≤ i ≤ j ≤ k satisfy | d ( W i , W j ) − d ( V i , V j ) | ≤ η . The ab o v e count ing lemma shows that if there is any mapp ing φ : [ h ] → [ k ] suc h that d ( W φ ( i ) , W φ ( j ) ) > η for ( i, j ) an edge of H and d ( W φ ( i ) , W φ ( j ) ) < 1 − η for i, j distinct and nonadjacent in H , then G con tains at least  η 4  ( h 2 ) | W 1 | × · · · × | W h | ≥ δn h induced copies of H . Hence, no su c h mapping φ exists. That is, for eve ry mapping φ : [ h ] → [ k ], there is an edge ( i, j ) for which d ( W φ ( i ) , W φ ( j ) ) ≤ η or distinct i, j that are nonadjacen t in H with d ( W φ ( i ) , W φ ( j ) ) ≥ 1 − η . F or eac h pair ( W i , W j ) f or which d ( W i , W j ) ≤ η , delete all edges b etw een V i and V j , and for eac h pair ( W i , W j ) for whic h d ( W i , W j ) ≥ 1 − η , add all edges b et wee n V i and V j . Let G ′ b e this mod iﬁcation of G . By the previous paragraph, there are no in duced copies of H in G ′ . W e hav e left to s ho w that few edge mo diﬁcations are made in obtaining G ′ from G . If a pair ( W i , W j ) for whic h edges w ere mo diﬁed satisﬁes | d ( W i , W j ) − d ( V i , V j ) | ≤ η , then the den sit y b etw een the t wo sets was only changed by at most 2 η . T h e n um b er of 1 ≤ i ≤ j ≤ k for w hic h | d ( W i , W j ) − d ( V i , V j ) | > η is at most η k 2 . Since V 1 , . . . , V k is an equipartition in to nonempt y parts, at m ost η k 2 · 4  n k  2 = 4 ηn 2 46 edges are c hanged b etw een such pairs. In total at most 2 η  n 2  + 4 ηn 2 ≤ 5 η n 2 < ǫn 2 edges w ere c hanged in order to obtain G ′ from G . F r om Lemma 5.11, w e hav e δ ′ = 1 8 S 2 , where S = S ( α, s, f ) is the fu nction from Lemma 5.10 with α = η 6 4 , s = 2 η − 1 and f ( k ) = 1 4 h η h . F rom Lemma 5.10, S ( α, s , f ) will be at most a to wer in h whose heigh t is p r op ortional to η − 6 . T herefore, by the c hoice of η and δ in the ab o ve pro of of the induced graph remo v al lemma, w e indeed get the desired to w er-t yp e b ound. This also completes the pro of of Theorem 1.3 . ✷ 6 Regular appro ximation lemma In th is s ection we show how to deriv e th e regular appr o ximation lemma from T ao’s regularit y lemma, as discussed in Sub section 1.5. Th e key lemma is Lemma 6.1 , whic h shows ho w to tur n a b ipartite graph in to a regular pair b y c h anging some edges according to a w eak regular partition. W e use the notation x = y ± ǫ to denote the f act that y − ǫ ≤ x ≤ y + ǫ . It will b e helpful to use the Ho eﬀding-Azuma inequalit y for concen tration of measur e. Sa y that a random v ariable X ( ω ) on an n -dimensional pro du ct space Ω = Q n i =1 Ω i is Lipschitz if c h anging ω in an y single co ordin ate aﬀects th e v alue of X ( ω ) by at most one. T he Ho eﬀding-Azuma inequalit y (see, e.g., [10]) pro vides concen tration for these distributions. Theorem 6.1 (Ho eﬀding-Azuma Inequality) L et X b e a Lipschitz r andom variable on an n - dimensional pr o duct sp ac e. Then for any t ≥ 0 , P [ | X − E [ X ] | > t ] ≤ 2 exp  − t 2 2 n  . F or a bipartite graph across p arts A and B , and partitions A : A = A 1 ∪ . . . ∪ A r and B : B = B 1 ∪ . . . ∪ B s , let q ( A, B ) = d 2 ( A, B ) and q ( A , B ) = P i,j | A i || B j | | A || B | d 2 ( A i , B j ) b e th e mean s q u are density across the partitions A and B . Lemma 6.1 L et 0 < δ < 1 . Supp ose A, B ar e disjoint v ertex subsets of a gr aph with | A | ≥ | B | > 8 δ − 2 and d ( A, B ) = η . Supp ose further that A : A = A 1 ∪ . . . ∪ A r and B : B = B 1 ∪ . . . ∪ B s form a we ak δ -r e gular p artition of the p air ( A, B ) , i.e. , for al l S ⊂ A and T ⊂ B , we have       r X i =1 s X j = 1 | A i ∩ S || B j ∩ T | d ( A i , B j ) − d ( S, T ) | S || T |       ≤ δ | A || B | . Then, one c an add or r emove at most  δ + ( q ( A , B ) − q ( A, B )) 1 / 2  | A || B | e dges acr oss ( A, B ) and thus turn it into a 2 δ 1 / 3 -r e gular p air satisfying d ( A, B ) = η ± δ . Pro of: Let α i,j = d ( A i , B j ) − d ( A, B ). If α i,j ≥ 0, w e delete eac h of the edges connecting A i and B j indep end en tly with p robabilit y α i,j d ( A i ,B j ) . If α i,j < 0, w e add ea c h of the nonedges b et wee n A i and B j 47 with probabilit y − α i,j 1 − d ( A i ,B j ) . Clearly the exp ected v alue of d ( A, B ) afte r these mo diﬁ cations is η . By the Ho eﬀd ing-Azuma inequality , th e p robabilit y that the n ew dens ity deviates from η by m ore than δ is at most 2 exp  − ( δ | A || B | ) 2 2 | A ||| B |  = 2 exp  − δ 2 | A || B | / 2  < 1 / 4 . Also, th e exp ected n umb er of edges c h anged is X i,j | α i,j || A i || B j | = X i,j | d ( A i , B j ) − d ( A, B ) || A i || B j | = | A || B | X i,j | d ( A i , B j ) − d ( A, B ) | p i q j ≤ | A || B |   X i,j ( d ( A i , B j ) − d ( A, B )) 2 p i q j   1 / 2 = | A || B | ( q ( A , B ) − q ( A, B )) 1 / 2 , where p i = | A i | / | A | and q j = | B j | / | B | and in the inequ alit y we used the Cauc hy-Sc h w arz inequalit y . By the Ho eﬀding-Azuma inequalit y , the pr obabilit y that the n umber of edges c hanged deviates b y more than δ | A || B | from its exp ected v alue is at most 2 exp  − ( δ | A || B | ) 2 2 | A || B |  = 2 exp  − δ 2 | A || B | / 2  < 1 / 4 . Consider no w t wo subsets A ′ ⊂ A and B ′ ⊂ B . As ( A, B ) was initially w eak δ -regular, the exp ected v alue of e ( A ′ , B ′ ) diﬀers from η | A ′ || B ′ | by at most δ | A || B | . By the Ho eﬀding-Azuma inequalit y , we get that the probabilit y that e ( A ′ , B ′ ) d eviates from its exp ected v alue b y more than δ | A || B | is at most 2 exp  − ( δ | A || B | ) 2 2 | A ′ || B ′ |  ≤ 2 exp  − δ 2 | A || B | / 2  < 2 exp {− 4 | A |} ≤ 2 −| A |−| B | − 2 , where we use | A | ≥ | B | > 8 δ − 2 . As there are 2 | A | + | B | c hoices for ( A ′ , B ′ ), we get that with probabilit y at least 3 / 4, all p airs ( A ′ , B ′ ) are within 2 δ | A || B | edges of ha ving edge density η . T o recap, w e get that with probabilit y at least 1 / 4 w e made at most  δ + ( q ( A , B ) − q ( A, B )) 1 / 2  | A || B | ed ge mo d iﬁcations, d ( A, B ) = η ± δ and all subsets A ′ ⊂ A, B ′ ⊂ B are within 2 δ | A || B | edges from ha vin g edge density η . Hence, there is su c h a c hoice for these edge mo diﬁcations, and w e claim that this implies that the p air ( A, B ) is 2 δ 1 / 3 -regular. Indeed, otherwise there w ou ld b e A ′ ⊂ A , B ′ ⊂ B , with | A ′ | ≥ 2 δ 1 / 3 | A | , | B ′ | ≥ 2 δ 1 / 3 | B | , and | d ( A ′ , B ′ ) − d ( A, B ) | > 2 δ 1 / 3 , whic h implies that A ′ , B ′ diﬀers by at least 2 δ 1 / 3 | A ′ || B ′ | ≥ (2 δ 1 / 3 ) 3 | A || B | = 8 δ | A || B | edges from ha ving edge densit y d ( A, B ), a con tradiction. This completes the pro of. ✷ W e next use Lemma 6.1 to deduce the regular appro ximation lemma from T ao’s regularit y lemma. Pro of: Let δ : N → (0 , 1) b e deﬁned by δ ( t ) = min  g ( t ) 3 32 t 2 , ǫ/ 2  . L et ǫ 0 = ( ǫ/ 2) 2 . Let T 0 = T 0 ( δ , ǫ 0 , s ) b e the b ound on the n u m b er of parts in T ao’s regularit y lemma and T = 16 T 0 /δ ( T 0 ) 2 . I f the num b er n of vertic es of the graph G s atisﬁes n ≤ T , then we can partition G into parts of size one, and the desired conclusion is satisﬁed in this case. Hence, w e m ay assu me n > T . By T ao’s regularit y lemma, the graph G h as an equitable v ertex partition P into t ≥ s p arts and an equitable v ertex reﬁnement Q into at most T 0 parts wh ic h is weak δ ( t )-regular suc h that q ( Q ) ≤ q ( P ) + ǫ 0 . 48 F or eac h pair of parts ( A, B ) of partition P , let A and B den ote th e partitions of A and B give n by partition Q . Since Q is a weak δ ( t )-regular partition, and A and B ha ve cardinalit y at least ⌊ n t ⌋ ≥ n 2 t , then the partitions A and B form a we ak 4 t 2 δ ( t )-regular partitio n. Note that 4 t 2 δ ( t ) ≤ g ( t ) 3 8 . Since | A | , | B | ≥ n 2 t > 8 /δ ( t ) 2 , w e ma y apply Lemma 6.1 to the graph b etw een A and B . Th at is, we ma y change at most  δ ( t ) + ( q ( A , B ) − q ( A, B )) 1 / 2  | A || B | ed ges across A and B and, in so doing, mak e ( A, B ) a g ( t )-regular pair, where we used that g ( t ) = 2  g ( t ) 3 8  1 / 3 . In total, the n umber of edges w e change to obtain a graph G ′ whic h is g -regular with resp ect to p artition P is at most X A,B ∈ P  δ ( t ) + ( q ( A , B ) − q ( A, B )) 1 / 2  | A || B | ≤ ( δ ( t ) + ǫ 1 / 2 0 ) n 2 ≤ ǫn 2 , where we used Jen sen’s inequalit y for the conca v e function h ( x ) = x 1 / 2 , th e in equ alit y q ( Q ) ≤ q ( P ) + ǫ 0 , and the b ounds δ ( t ) ≤ ǫ/ 2, ǫ 0 = ( ǫ/ 2) 2 . T o complete the pro of, we recall that the num b er of parts in partition P is at lea st s and at most T 0 = T 0 ( δ , ǫ 0 , s ). ✷ 7 F rieze-Kannan w eak regularit y lemma In this section w e prov e Th eorem 1.4 which p ro vid es a lo w er b ound on the weak regularit y lemma. F or a verte x partition P : V = V 1 ∪ . . . ∪ V k of a graph G = ( V , E ), let f P ( A, B ) = f G P ( A, B ) = e ( A, B ) − X 1 ≤ i,j ≤ k d ( V i , V j ) | A ∩ V i || B ∩ V j | , whic h is the diﬀerence b etw een the num b er of edges b et we en A and B and the exp ected num b er of edges based on the densities across the pairs of parts of the partition and the intersecti on sizes of A and B with the p arts. W e call a partitio n P of the v er tex set of a graph G = ( V , E ) we ak ǫ -r e gular if it satisﬁes | f P ( A, B ) | ≤ ǫ | V | 2 for a ll A, B ∈ V . Recall that the w eak r egularit y lemma stat es that for eac h ǫ > 0 there is a p ositiv e in teger k ( ǫ ) su c h that ev ery graph has an equ itable w eak ǫ -partition into at most k ( ǫ ) parts. Moreo v er, one ma y tak e k ( ǫ ) = 2 O ( ǫ − 2 ) . W e will p ro ve that the num b er of parts r equired in the w eak regularit y lemma satisﬁes k ( ǫ ) = 2 Ω( ǫ − 2 ) , th us mat c h ing the upp er b ound. The follo wing simple lemma of F rieze and Kannan (see Lemm a 7(a) of [20]) sh o ws that the notion of w eak r egularit y is robu s t. Lemma 7.1 If a p artition is we ak ǫ -r e gular, then any r eﬁnement of it is we ak 2 ǫ -r e gular. The robu stness of w eak regularit y describ ed by Lemma 7.1 is n ot shared b y the usual notion of regular partition. F or example, f or any ﬁ xed ǫ > 0 and p ositiv e integ er t , almost su rely an y partition into t parts of a un iform rand om graph on suﬃcien tly man y v ertices is ǫ -regular, while an y partition of the 49 v ertex set in to parts of size 2 is not ( ǫ, δ , η )-regular w ith ǫ = 1, δ = η = 1 / 2. Th is is b ecause al most surely in any suc h partitio n, b et wee n most pairs of p arts of size 2, there will b e at lea st one edge and at least one nonedge. What we will actually prov e is the s tr onger result that an y weak ǫ -regular p artition must ha v e 2 Ω( ǫ − 2 ) parts, whether it is equ itable or not. The corresp onding regularit y lemma, whic h is an immediate corollary of the usual weak regularit y lemma, is the follo win g. Lemma 7.2 F or e ach ǫ > 0 ther e is a p ositive inte ger k ∗ ( ǫ ) such that every gr aph G = ( V , E ) has a vertex p artition P with at most k ∗ ( ǫ ) p arts which is we ak ǫ -r e gular. In the other direction, the equitable version of the wea k regularit y lemma also follo ws from L emma 7.2. This is b ecause of the r obustness prop erty d iscu ssed in Lemma 7.1 ab o ve, that is, an y reﬁnement of a wea k ǫ -regular partition is a 2 ǫ -regular p artition. By arb itrarily reﬁning a not necessarily equitable partition in to an equitable partition (except f or a small fraction of ve rtices, w hic h we distribu te ev enly amongst the other parts), we get an equitable wea k 3 ǫ -partition w h ose num b er of parts is only a factor p olynomial in ǫ − 1 larger. In order to p r o ve the lo w er b ound for w eak regularit y , we will need to p erf orm some fur th er r eductions. W e ﬁrst stat e a bipartite v ariant which can easily b e sho wn to b e equ iv alent to Lemma 7.2. F or a bipartite graph G = ( U, V , E ) with | U | = | V | = n , p artitions P 1 : U = U 1 ∪ . . . ∪ U k and P 2 : V = V 1 ∪ . . . ∪ V k ′ , and v ertex subsets A ⊂ U and B ⊂ V , let f P 1 ,P 2 ( A, B ) = f G P 1 ,P 2 ( A, B ) = e ( A, B ) − k X i =1 k ′ X j = 1 d ( U i , V j ) | A ∩ U i || B ∩ V j | . W e call the pair of partitions P 1 , P 2 we ak ǫ -r e gular with resp ect to th e bip artite graph G if | f P 1 ,P 2 ( A, B ) | ≤ ǫn 2 for all A ⊂ U and B ⊂ V . Lemma 7.3 F or e ach ǫ > 0 ther e is a p ositive inte ger k ′ ( ǫ ) such that eve ry bip artite gr aph G = ( U, V , E ) with p arts of e qual size has p artitions P 1 of U and P 2 of V e ach with at most k ′ ( ǫ ) p arts which form a we ak ǫ - r e gular p artition. T o prov e Th eorem 1.4, it suﬃces to s h o w k ′ ( ǫ ) = 2 Ω( ǫ − 2 ) . Indeed, th is follo ws f rom the b ound k ′ ( ǫ ) ≤ k ∗ ( ǫ/ 2). This in equalit y follo ws fr om ﬁ rst considering a single weak ǫ/ 2-regular partition P for the bipartite graph G in to at most k ∗ ( ǫ/ 2) parts, and then reﬁning it in to a partition P ′ with at most 2 k ∗ ( ǫ/ 2) parts b ased on the in tersections of the parts of P w ith U and V . By Lemma 7.1, P ′ is a we ak ǫ -regular partition. Letting P 1 b e the parts of P ′ in U and P 2 b e the parts of P ′ in V , the p air P 1 , P 2 form a weak ǫ -regular p artition, eac h with at most k ∗ ( ǫ/ 2) p arts, so that k ′ ( ǫ ) ≤ k ∗ ( ǫ/ 2). T o get a lo wer b oun d for the weak regularit y lemma, we do not need to sho w the other direction of th e equiv alence b et w een the weak regularit y lemma and Lemma 7.3 , that Lemma 7.3 imp lies the 50 w eak r egularit y lemma. Ho wev er, this is rather simple, so w e sketc h it here. F rom a graph G w e ca n consider the bipartite d ouble co v er of G , which is the tensor pro d uct of G with K 2 . Applyin g Lemma 7.3, we get a pair P 1 , P 2 of partitions of V ( G ) whic h form a w eak ǫ/ 2-regular partition with resp ect to the bipartite double co v er of G . Reﬁ n ing the t w o partitions P 1 , P 2 of V ( G ), w e get by Lemma 7.1 a wea k ǫ -regular partition f or G , thus establishing k ∗ ( ǫ ) ≤ k ′ ( ǫ/ 2) 2 . The follo wing tec hnical lemma will allo w u s to construct a w eigh ted graph rather than a graph. A similar idea is pr esen t in the lo wer b ound construction of Go wers [22] for Szemer´ edi’s regularit y lemma. Let W b e a [0 , 1]-v alued n × n matrix. W e view W as a w eigh ted graph with parts U and V , where U and V denote the set of columns and ro ws, resp ectiv ely , of W . Let e W ( A, B ) = P a ∈ A,b ∈ B W ( a, b ) and d W ( A, B ) = e W ( A,B ) | A || B | . Lemma 7.4 L et M b e an n × n matrix with entries in the interval [0 , 1] . L et G = ( U, V , E ) b e a bip artite r andom gr aph with | U | = | V | = n and e dges chosen indep endently given by M and let θ = 4 n − 1 / 2 . With pr ob ability at le ast 1 − e − 4 n , we have | e M ( A, B ) − e G ( A, B ) | ≤ θ n 2 for every p air of sets A ⊂ U , B ⊂ V . Pro of: Giv en t w o vertic es u ∈ U and v ∈ V , let a ( u, v ) b e the rand om v ariable G ( u, v ) − M ( u, v ) (where G has b een iden tiﬁed with its adj acency matrix). T he mean of a ( u, v ) is zero for all u, v and the mod ulus of a ( u, v ) is at most 1. Hence, b y the Ho eﬀding-Azuma inequalit y (Theorem 6.1), giv en t wo sets A ⊂ U and B ⊂ V , the p r obabilit y that       X ( u,v ) ∈ A × B a ( u, v )       ≥ θ n 2 is at most 2 exp  − ( θ n 2 ) 2 / (2 | A || B | )  ≤ 2 exp {− 8 n } . Summ ing o v er all A ⊂ U and B ⊂ V , the probabilit y that there are subsets A ⊂ U and B ⊂ V with | e G ( A, B ) − e M ( A, B ) | ≥ θ n 2 is at most 2 2 n · 2 e − 8 n ≤ e − 4 n . ✷ F or partitions P 1 : U = U 1 ∪ . . . ∪ U k and P 2 : V = V 1 ∪ . . . ∪ V k ′ , let f P 1 ,P 2 ( A, B ) = e W ( A, B ) − k X i =1 k ′ X j = 1 d W ( U i , V j ) | U i ∩ A || V j ∩ B | . W e say that partitions P 1 , P 2 form a we ak ǫ -r e gular p artition of W if | f P 1 ,P 2 ( A, B ) | ≤ ǫn 2 for all su bsets A ⊂ U and B ⊂ V . Corollary 7.1 Supp ose W = ( U, V , E ) is an e dge- weig hte d gr aph with weights in [0 , 1] and | U | = | V | = n . L et G = ( U, V , E ) b e a bip artite r andom gr aph with | U | = | V | = n and e dges chosen indep endently g iven by W and let θ = 4 n − 1 / 2 . With pr ob ability at le ast 1 − e − 4 n , every p air of p artitions P 1 : U = U 1 ∪ . . . ∪ U k and P 2 : V = V 1 ∪ . . . ∪ V k ′ which form a we ak ǫ -p artition for G also form a we ak ( ǫ + 2 θ ) -r e gular p artition for W . 51 Pro of: By Lemma 7.4, with pr obabilit y at least 1 − e − 4 n , we ha v e | e G ( A, B ) − e W ( A, B ) | ≤ θ n 2 for ev ery pair of v ertex subsets A ⊂ U and B ⊂ V . Su pp ose this in deed holds. F or graph G , w e h a ve f G P 1 ,P 2 ( A, B ) = e G ( A, B ) − k X i =1 k ′ X j = 1 d G ( U i , V j ) | A ∩ U i || B ∩ V j | . The ﬁrst term is within θ n 2 of e W ( A, B ). The s econd term is the a verage of e G ( A ′ , B ′ ) ov er all sub sets A ′ ⊂ U and B ′ ⊂ V with | A ′ ∩ U i | = | A ∩ U i | for all i and | B ′ ∩ V j | = | B ∩ V j | for all j , and hence is within θ n 2 of the corresp ondin g av erage of e W ( A ′ , B ′ ) o ver all of the same p airs ( A ′ , B ′ ). By the triangle inequalit y , for W , w e get that for all A ⊂ U an d B ⊂ V , w e ha ve | f W P 1 ,P 2 ( A, B ) | ≤ | f G P 1 ,P 2 ( A, B ) | + 2 θ n 2 ≤ ( ǫ + 2 θ ) n 2 . Hence, P 1 , P 2 also form a w eak ( ǫ + 2 θ )-regular partition f or W . ✷ F r om C orollary 7.1 and the previous remarks, to obtain the desired lo w er b ound in Th eorem 1.4 on the num b er of parts in the weak regularit y lemma it suﬃces to pro v e a lo wer b ound of the form 2 Ω( ǫ − 2 ) on th e n umber of parts k 0 ( ǫ ) in the f ollo wing w eak regularit y lemma for w eigh ted bipartite graphs. Lemma 7.5 F or e ach ǫ > 0 ther e is a p ositive inte ger k 0 ( ǫ ) such that every e dge-weig hte d bip artite gr aph G = ( U, V , E ) with weights i n [0 , 1] and p arts of e qual size has p artitions P 1 of U and P 2 of V e ach with at most k 0 ( ǫ ) p arts which form a we ak ǫ -r e gular p artition. Lemma 7.5 is also known as the we ak matrix r egularit y lemma. This is b ecause it provides, for an y n × n matrix with en tries in [0 , 1], partitions of the ro w s and columns in to a b ound ed n umber of parts, suc h that the sum of the ent ries in any submatrix (wh ic h is th e pro d uct of a set of ro w s and column s) is within ǫn 2 of what is exp ected b ased on the intersectio ns w ith the p arts and the den sit y b et wee n the p arts. Our goal is to constr u ct a bipartite graph G w ith edge weigh ts in [0 , 1] whic h provides a low er b ound of the form k 0 ( ǫ ) = 2 Ω( ǫ − 2 ) . Supp ose 0 < ǫ ≤ 2 − 50 . Consider th e follo wing we igh ted bipartite graph G . The graph has parts U and V eac h of order n = 2 2 − 45 ǫ − 2 . Let r = 2 − 40 ǫ − 2 and α = 2 14 ǫ . Consid er , for 1 ≤ i ≤ r , equita ble partitions U = U i 0 ∪ U i 1 and V = V i 0 ∪ V i 1 pic ked u niformly and indep en den tly at random. F or v ertices u ∈ U and v ∈ V , let s ( u, v ) b e the n umber of i ∈ [ r ] f or whic h there is j ∈ { 0 , 1 } suc h that u ∈ U i j and v ∈ V i j , and t ( u, v ) b e the num b er of i ∈ [ r ] for wh ic h ther e is j ∈ { 0 , 1 } such that u ∈ U i j and v ∈ V i 1 − j , so that s ( u, v ) + t ( u, v ) = r . Let W ( u, v ) = 1 2 + ( s ( u, v ) − t ( u, v )) α . W e deﬁne the w eigh t w ( u, v ) b et wee n u and v as follo ws. If 0 ≤ W ( u, v ) ≤ 1, then w ( u, v ) = W ( u, v ), if W ( u, v ) < 0, then w ( u, v ) = 0, and if W ( u, v ) > 1, then w ( u, v ) = 1. Call a pair ( u, v ) ∈ U × V extr eme if | W ( u, v ) − 1 / 2 | > 1 / 4, and a v ertex u ∈ U nic e if it is in at most n/ 8 pairs ( u, v ) with v ∈ V whic h are extreme. Lemma 7.6 With pr ob ability at le ast 3 / 4 , al l but at mo st e − 100 n vertic e s of U ar e nic e. Pro of: Fix a pair ( u, v ) ∈ U × V . The ev ent ( u, v ) is extreme is the same as | s ( u, v ) − t ( u, v ) | α > 1 / 4, or equiv alen tly that | s ( u, v ) − r / 2 | > 1 8 α . The num b er s ( u, v ) is a sum of r indep en den t v ariables, with 52 v alues 0 or 1 eac h with p robabilit y 1 / 2 , and hence follo ws a b inomial distribution with mean r / 2. By Chernoﬀ ’s b ound (1), the probability that | s ( u, v ) − r / 2 | > 1 8 α is less than 2 e − 2(1 / (8 α )) 2 /r = 2 e − 2 7 . Hence, b y linearit y of exp ectation, the exp ected n u m b er of extreme pairs ( u, v ) ∈ U × V is less than 2 e − 2 7 n 2 . Therefore, by Mark o v’s inequalit y , the probab ility that th er e are at least 8 e − 2 7 n 2 extreme pairs ( u, v ) is less than 1 / 4. Since any n ice v ertex is con tained in at most n/ 8 extreme pairs, we see that with probabilit y at least 3 / 4, all but a t m ost 64 e − 2 7 n ≤ e − 100 n vertic es in U are nice. ✷ F or h ∈ [ r ], w e let s h ( u, v ) den ote th e n u m b er of i ∈ [ r ] \ { h } for whic h there is j ∈ { 0 , 1 } su c h th at u ∈ U i j and v ∈ V i j , and t h ( u, v ) b e th e n umb er of i ∈ [ r ] \ { h } for wh ic h there is j ∈ { 0 , 1 } su c h that u ∈ U i j and v ∈ V i 1 − j , so that s h ( u, v ) + t h ( u, v ) = r − 1. Let W h ( u, v ) = 1 2 + ( s h ( u, v ) − t h ( u, v )) α . As ab ov e, w e deﬁn e the we igh t w h ( u, v ) b y w h ( u, v ) = W h ( u, v ) if 0 ≤ W h ( u, v ) ≤ 1, w h ( u, v ) = 0 if W h ( u, v ) < 0, and w h ( u, v ) = 1 if W h ( u, v ) > 1. Lemma 7.7 Supp ose u ∈ U h j and we ar e given | w h ( u, v ) − 1 / 2 | ≤ 1 / 2 − α for at le ast 7 8 n vertic es v ∈ V , and we do not yet know the p artition V = V h 0 ∪ V h 1 . Then the pr ob ability that d w ( u, V h j ) − d w ( u, V h 1 − j ) ≥ α/ 2 is at le ast 1 − 1 4 rn . Pro of: Consider the ev en t E that X v ∈ V h 1 − j w h ( u, v ) − X v ∈ V h j w h ( u, v ) ≥ αn / 4 . Note that the exp ected v alue of the left hand side is 0. Recall that a hyp er geometric d istr ibution is at least as concent rated as the sum of indep enden t random v ariables with the same v alues (for a pro of, see Section 6 of [25]). By the Hoeﬀding-Azuma inequalit y (Theorem 6.1), the pr obabilit y of ev ent E is at most 2 exp  − ( αn/ 8) 2 2 n  = 2 exp  − 2 − 7 α 2 n  ≤ 1 4 r n , where in the last inequ alit y w e use 0 < ǫ ≤ 2 − 50 , r = 2 − 40 ǫ − 2 , n = 2 2 − 45 ǫ − 2 , and α = 2 14 ǫ . F or a ﬁxed u , if | w h ( u, v ) − 1 2 | ≤ 1 2 − α , then w ( u, v ) = w h ( u, v ) + α if v is in V h j and w ( u, v ) = w h ( u, v ) − α if v is in V h 1 − j . F or all v , w ( u, v ) is within α of w h ( u, v ). Therefore, letting w ( u, S ) = P s ∈ S w ( u, s ), w e see, since | w h ( u, v ) − 1 2 | ≤ 1 2 − α for at least 7 8 n vertic es of v , th at w ( u, V h j ) − w ( u, V h 1 − j ) ≥ 7 8 αn − 1 8 αn + w h ( u, V h j ) − w h ( u, V h 1 − j ) ≥ 3 4 αn − 1 4 αn = α 2 n. The resu lt follo ws. ✷ Call a nice v ertex u ∈ U very nic e if for eac h h ∈ [ r ] and j ∈ { 0 , 1 } with u ∈ U h j , d ( u, V h j ) − d ( u, V h 1 − j ) ≥ α/ 2 . Corollary 7.2 With pr ob ability at le ast 3 / 4 , every nic e vertex u is very nic e. 53 Pro of: Giv en u is nice, then for eac h h ∈ [ r ], w e must hav e | W h ( u, v ) − 1 / 2 | ≤ 1 / 2 − α for all but at most 7 8 n v ertices v ∈ V . The probabilit y that there is a v ertex which is nice but not v ery nice is b y Lemma 7.7 at most r n · 1 4 rn ≤ 1 / 4, whic h co mpletes the pro of. ✷ F r om Lemma 7.6 and Corollary 7.2, we hav e the follo win g corollary . Corollary 7.3 With pr ob ability at le ast 1 / 2 , the gr aph G has the fol lowing pr op e rties. • The numb er of vertic e s in U which ar e not nic e is at most e − 100 n . • Every nic e vertex is very nic e. Consider the random bip artite graph B = B ( n, r ) w ith vertex parts U and [ r ] wh ere i ∈ [ r ] is adjacen t to u ∈ U if u ∈ U i 0 . By Lemma 2.1 with µ = 1 / 4, as r ≥ 32 log n , w e ha v e the follo wing prop osition. Corollary 7.4 With pr ob ability at le ast 3 / 4 , for e ach p air u, u ′ ∈ U , the numb er of i for which u and u ′ b oth b elong to U i j for some j ∈ { 0 , 1 } is less than 3 4 r . Hence, with pr obabilit y at least 1 / 4, the graph G satisﬁes th e p r op erties in Corollaries 7.3 and 7.4. Fix su c h a graph G . Theorem 7.1 No p artitions P 1 : U 1 ∪ . . . ∪ U k of U and P 2 : V 1 ∪ . . . ∪ V k ′ of V with k ≤ n/ 2 form a we ak ǫ -r e gular p artition. As n/ 2 ≥ 2 2 − 46 ǫ − 2 , we ther efor e have k 0 ( ǫ ) > 2 2 − 46 ǫ − 2 for 0 < ǫ ≤ 2 − 50 . Theorem 7. 1 gives a lo w er b ound on the num b er k 0 ( ǫ ) of parts for the w eak matrix regularit y lemma (Lemma 7.5) with appro ximation ǫ . Before w e prov e th is theorem, we remark that it has n o restriction on the n u m b er of parts of p artition P 2 , and further sho w s th at P 1 has to b e almost th e ﬁnest partition (partition in to sin gletons) to obtain a pair of partitions wh ic h are w eak ǫ -regular. Pro of: Supp ose f or contradictio n that the p artitions P 1 and P 2 are weak ǫ -regular. T hat is, | f P 1 ,P 2 ( A, B ) | ≤ ǫn 2 for all subsets A ⊂ U and B ⊂ V . Fix for no w U t with | U t | ≥ 2. Call th e pair ( i, t ) ∈ [ r ] × [ k ] useful if | U t ∩ U i j | ≥ | U t | / 32 for j ∈ { 0 , 1 } . Let M t b e the n u m b er of i ∈ [ r ] for wh ic h the pair ( i, t ) is us efu l. The sum S t = X i ∈ [ r ] | U t ∩ U i 0 || U t ∩ U i 1 | ≤ r | U t | 2 / 32 + M t | U t | 2 / 4 is p recisely th e num b er of triples u, u ′ , i with u, u ′ distinct elemen ts of U t and i ∈ [ r ] f or w h ic h u and u ′ are not in the same set in the partition U = U i 0 ∪ U i 1 . By Corollary 7.4, the su m S t is at lea st 1 4 r  | U t | 2  ≥ | U t | 2 r / 16. Hence, r | U t | 2 / 32 + M t | U t | 2 / 4 ≥ S t ≥ | U t | 2 r / 16 . W e th us h a ve M t ≥ r / 8. 54 Since M t ≥ r / 8 f or eac h t for wh ic h | U t | ≥ 2 and there are at most k parts in partition P 1 of ord er 1, there is an i for whic h partition i satisﬁes that at least ( n − k ) / 8 ≥ n/ 16 vertice s u ∈ U are in U t with the p air ( i, t ) useful. Fix s uc h an i . F or eac h t for w h ic h ( i, t ) is usefu l and all bu t at m ost | U t | / 64 v ertices in U t are nice, for j ∈ { 0 , 1 } , let U t,j b e a subset of U t ∩ U i j of cardinalit y exactly ⌈| U t | / 64 ⌉ , and A j denote the u n ion of all suc h U t,j . Recall from Corollary 7.3 that there are at most e − 100 n v ertices in U wh ich are not nice. Hence, there are at most 64 · e − 100 n v ertices in U wh ic h b elong to a U t for whic h the p air ( i, t ) is useful but there are at least | U t | / 64 v ertices in U t whic h are n ot nice. Th us, the num b er of v ertices in U whic h b elong to a U t for wh ic h ( i, t ) is useful and there are at most | U t | / 64 vertic es in U t whic h are not nice is at lea st n 16 − 64 e − 100 n > n 32 . W e th us h a ve | A 0 | = | A 1 | > n 32 / 64 = 2 − 11 n . Note that, b y co nstruction, w e hav e f or eac h t ∈ [ k ], ℓ ∈ [ k ′ ] and T ⊂ V , | A 0 ∩ U t || T ∩ V ℓ | d ( U t , V ℓ ) = | A 1 ∩ U t || T ∩ V ℓ | d ( U t , V ℓ ) . Th us, if the partitions P 1 , P 2 form a w eak ǫ -regular partition, we w ould h a ve to h a ve | e ( A 0 , V i j ) − e ( A 1 , V i j ) | ≤ 2 ǫn 2 . (29) for j ∈ { 0 , 1 } . Ho w ev er, as eac h u ∈ A 0 is in U i 0 and is v ery nice, w e hav e d ( A 0 , V i 0 ) − d ( A 0 , V i 1 ) ≥ α/ 2 . Since | A 0 | > 2 − 11 n and | V i j | = n/ 2 for j ∈ { 0 , 1 } , w e ha ve e ( A 0 , V i 0 ) − e ( A 0 , V i 1 ) > 2 − 13 αn 2 . Similarly , e ( A 1 , V i 1 ) − e ( A 1 , V i 0 ) > 2 − 13 αn 2 . Adding the previous t wo inequ alities, we ha ve e ( A 0 , V i 0 ) − e ( A 1 , V i 0 ) + e ( A 1 , V i 1 ) − e ( A 0 , V i 1 ) > 2 − 12 αn 2 . (30) But, by (29) for j ∈ { 0 , 1 } , the left hand side of (30 ) is at most 4 ǫn 2 in mo d ulus, contradicti ng the ab o v e inequalit y and α = 2 14 ǫ . This completes the pro of. ✷ Remark: While Th eorem 7.1 pro vides for eac h ǫ only one graph w hic h requir es at least 2 Ω( ǫ − 2 ) parts in an y weak ǫ -regular pair of partitions, it is a simple exercise to mo dify the pr o of to show that all blo w -u ps of G also satisfy this prop erty , thus obtaining an inﬁn ite family of suc h graphs. F or a graph G on n vertices and a p ositiv e integ er t , the blow-up G ( t ) of G is the graph on nt v ertices obtained b y replacing eac h verte x u b y an in dep endent set I u , and a vertex in I u is adjacen t to a v ertex in I v if and only if u and v are adjace n t. 55 8 Concluding remarks • W eak regularity lemmas without irregular pairs While pr o ving his famous th eorem on arithmetic progressions in dens e su b sets of the integ ers, Szemer ´ edi [39] actually devel op ed a regularit y lemma w hic h is weak er than w hat is now com- monly kno wn as Szemer´ edi’s regularit y lemma [40]. The follo win g v ersion is a strengthening of th e original version, as it guaran tees that all pairs, ins tead of all b ut an ǫ -fraction of pairs, under consideration are ǫ -r egular. The key extra ingredient is an application of Lemma 5.6 to redistribute the small fraction of v ertices whic h are not in regular sets. Lemma 8.1 F or e ach 0 < ǫ < 1 / 2 ther e ar e inte gers k = k ( ǫ ) and K = K ( ǫ ) such that the fol lowing holds. F or every gr aph G = ( V , E ) , ther e is an e quitable p artition V = V 1 ∪ . . . ∪ V t into at most k p arts such tha t for e ach i , 1 ≤ i ≤ t , ther e is a p artition V = V i 1 ∪ . . . ∪ V ij i , with j i ≤ K , suc h that for al l 1 ≤ i ≤ t and 1 ≤ j ≤ j i the p air ( V i , V ij ) is ǫ -r e gular. F u rthermor e, k ( ǫ ) = 2 ǫ − C and K ( ǫ ) = O ( ǫ − 1 ) , wher e C is an absolute c onstant. Szemer ´ edi [39] originally ga ve a triple exp on ential upp er b ound on the n umber of parts in the original regularit y lemma, whereas it is no w known (see [33]) that the correct b oun d is single exp onent ial. Through iterativ e applications, th e original regularit y lemma w as used by Ruzsa and S zemer ´ edi [36] to resolv e the (6 , 3)-problem, and by Szemer ´ edi [38] to establish the u p p er b ound on the Ramsey-T ur´ an problem for K 4 . It is a relativ ely simple exercise to sho w that Szemer ´ edi’s original r egularit y lemma implies the F r ieze-Kannan weak regularit y lemma, bu t with a b ound that is one exp onential wo rse than the tigh t b ound established in the previous section. This can b e acc omplished by sho wing th at th e common reﬁn emen t of the partitions in the original regularit y le mma sati sﬁes the F rieze-Kannan weak regularit y le mma. There are a n umb er o f notable prop erties of Lemma 8.1. First, all pairs ( V i , V ij ) under consid- eration in Lemma 8.1 are regular. In contrast, Theorem 1.1 sho w s that w e m u st allo w for many irregular pairs in Szemer ´ edi’s regularit y lemma. Second, the b ounds are m uch b etter than in Szemer ´ edi’s regularit y lemma. Th e b oun ds on the num b er of p arts is only single-exp onent ial, instead of th e to w er-t yp e b ou n d which app ears in the s tandard r egularit y lemma. F urthermore, eac h of the partitions of V ha v e at most K ( ǫ ) = O ( ǫ − 1 ) parts. Ind eed, this can b e established b y ﬁrst proving an y b ound on K ( ǫ ), and then using the follo wing additiv e p rop erty of regularit y to combine parts. Namely , applying Lemma 8.1 with an y b ound on K ( ǫ ) and w ith ǫ 2 / 4 in place of ǫ , and , for eac h i , partitioning V in to O ( ǫ − 1 ) p arts, eac h p art consisting o f the union of parts V ij for whic h d ( V i , V ij ) lies in an in terv al of length at most ǫ/ 2, the f ollo wing lemma sh ows that V i together with eac h part of the new partition f orms an ǫ -regular pair. Lemma 8.2 L et 0 < ǫ < 1 and α = ǫ 2 / 4 . Supp ose A, B 1 , . . . , B r ar e disjoint sets satisfying ( A, B i ) is α -r e gular for 1 ≤ i ≤ r and | d ( A, B i ) − d ( A, B j ) | ≤ ǫ/ 2 for 1 ≤ i, j ≤ r . Then, letting B = B 1 ∪ . . . ∪ B r , the p air ( A, B ) is ǫ -r e gular. 56 T o sa ve space, w e omit the details of ho w to pro ve Lemmas 8.1 and 8.2. Another inte resting consequence of Lemma 8.1 is that it implies that every graph on n v ertices has an ǫ -regular pair in which one p art is of size Ω( ǫn ) and the other p art h as size 2 − ǫ − O (1) n . It is w ell known (see [27]), that one can ﬁnd an ǫ -regular pair in whic h b oth p arts ha ve size 2 − ǫ − O (1) n . In some app lications, h aving one regular pair is su ﬃ cien t (see, e.g., [16], [24], [27]), and one obtai ns m uc h b etter b ound s than by applyin g Szemer ´ edi’s regularity lemma. In the other d irection, there are graph s (see Theorem 1.4 in [31] for a tigh t result) for which an y ǫ - regular p air h as a part of s ize 2 − ǫ − Ω(1) n . W e can nev ertheless guaran tee that one part is of size Ω( ǫn ). It seems like ly th at ha v in g such a large part in a r egular pair could b e useful in some applications. By iterativ e app lication of Lemma 8.1, one can also obtain a ve rsion of the Duk e-Lefmann-R¨ odl lemma suc h that all cylind ers in the partition are ǫ -regular. In fact, using Lemma 5.7, one can further guarant ee that all cylinders in the partition are strongly ǫ -regular, and the b oun d is still of constan t-to we r h eigh t. • A part irregular with no other part s In the pro of of Theorem 1.1, w e found a graph G su c h that for an y partition in to k parts there are at least θ k parts V i for which there are at least θ − 1 η k pairs ( V i , V j ) whic h are not ( ǫ, δ )-regular, where 0 < θ < 1 is a ﬁ x ed constan t. Is it p ossible to imp r o ve this resu lt so that all parts are in η k irregular pairs? The answ er is no, as Lemma 8.1 implies that any graph has an equitable partition into only 2 ǫ − O (1) parts in whic h one part is ǫ -regular with all the other p arts. • A single regular subset It w ould b e in teresting to determine tigh t b ound s for the size of the largest ǫ -regular subset whic h can b e found in ev ery graph . In Lemm a 5.2 , w e show ed that ev ery graph G = ( V , E ) must con tain an ǫ -regular sub set U of size at least 2 − ǫ − (10 /ǫ ) 4 | V | . On th e other h an d , a r esult of P en g, R¨ odl and Ruci ´ nski [31] implies that there are graph s G = ( V , E ) whic h con tain n o ǫ -regular subset of size ǫ cǫ − 1 | V | . W e conjecture that the correct b ound is single exp onen tial, though the p o w er may b e p olynomial in ǫ − 1 . • Equitable partitions and not necessarily equitable partitions In the regularity lemmas considered in this pap er, we often assume the p artitions we consider are equitable partitions. It is not diﬃcult to see that this assump tion is n on-essen tial and the b ound s do not c hange m uc h. Indeed, consider for example the follo wing v arian t of Szemer ´ edi’s regularit y lemma. Lemma 8.3 F or e ach ǫ, δ, η > 0 ther e is a p ositive inte ger M = M ( ǫ, δ, η ) for which the f ol lowing holds. Every gr aph G = ( V , E ) has a vertex p artition V = V 1 ∪ . . . ∪ V k with k ≤ M su c h that the sum of | V i || V j | over al l p airs ( V i , V j ) which ar e not ( ǫ, δ ) -r e gular is at most η | V | 2 . 57 The k ey d iﬀerence b et wee n this version of Szemer´ edi’s regularit y lemma and the usual statemen t is that the parts of the partition are not necessarily of equal order, and w e measure the regularit y of the partition by the sum of the pro d ucts of the sizes of th e p airs of parts that are ( ǫ, δ )-regular. Szemer ´ edi’s regularit y lemma clearly implies Lemma 8.3, as it fur ther sp eciﬁes th at the partition is an equitable partition, and the condition that the s u m of | V i || V j | o v er all pairs ( V i , V j ) which are not ( ǫ, δ )-regular is at most η | V | 2 is then essent ially the same as saying th at the num b er of pairs ( V i , V j ) whic h are not ( ǫ, δ )-regular is at most η k 2 . T o see that Lemma 8.3 implies Szemer ´ ed i’s regularit y lemma with similar b oun d s, ﬁr st apply Lemma 8.3, and then r andomly reﬁ n e eac h part V i = V i 0 ∪ V i 1 ∪ . . . ∪ V ij i in to parts of size m = 1 100 α 2 | V | / M , where α = m in( ǫ, δ, η ), except p ossibly V i 0 , whic h can ha v e size less than m a s not necessa rily eac h V i has ca rdinalit y a m ultiple of m . The to tal num b er of r emainin g v ertices, those in V 0 = S k i =1 V i 0 , is less than k m ≤ 1 100 α 2 | V | , as there are less than m remaining vertic es from eac h of the k parts V i . Redistribu ting the v ertices of V 0 ev enly among the parts of size m , w e get a new equitable partition where eac h p art has size b et w een m and at most (1 + 1 50 α 2 ) m . It is not hard to show using an a dditional application of the F rieze-Kannan w eak regularit y lemma, th at b ecause we randomly r eﬁned eac h part, almost surely for all pairs ( V h , V i ) wh ic h are ( ǫ, δ )-regular, all pairs ( V ha , V ij ) are (2 ǫ, 2 δ )-regular. That is, the regularit y b et ween pairs of parts is almost surely inherited b et ween large vertex sub s ets. It then easily follo ws that the equitable p artition is sim ilar b oth in the num b er of parts and the degree of r egularit y to the original partition fr om Lemma 8.3 . Because of this equiv alence, w e get similar lo w er b ou n ds for regularit y lemmas where the par- titions are n ot n ecessarily equitable partitio ns. F or example, for Lemma 8. 3, for some absolute constan ts ǫ, δ > 0, we get a b ound on M ( ǫ, δ , η ) wh ic h is a to w er of 2s of height Ω( η − 1 ). Simi- larly , in Theorem 1.2 and Corollary 1.1 giving lo wer b ounds on the strong regularit y lemma, the assumption that B is an equitable partition is not needed. Finally , as w e hav e already noted in Section 7, it is muc h easier to sho w that for the F rieze- Kannan w eak regularit y lemma w e do not need to assume that the p artition is equitable. This is a simple consequ en ce of the r obustness of w eak r egularit y under reﬁ nemen t. • Irregular pairs and half graphs A (generalized) half-graph has v ertex set A ∪ B with 2 n vertic es A = { a 1 , . . . , a n } and B = { b 1 , . . . , b n } , in which ( a i , b j ) is an edge if and only if i ≤ j (bu t the edges within A and B could b e arbitrary). As mentio ned in the introdu ction, an y partition of a large half-graph in to a co nstan t n u m b er of parts has some irregular pairs. Malliaris and Shelah [30] rece n tly sho w ed that for eac h ﬁxed k , eve ry graph on n vertices with no induced sub grap h wh ic h is a half-graph on 2 k v ertices h as an ǫ -regular partition with no irr egular parts and the n umber of parts is at most ǫ − c k , w h ere c k is single-exp onenti al in k . This sho ws that any construction forcing irr egular pairs in the regularity lemma, lik e that giv en in the pr o of of Theorem 1.1, must con tain large half-graphs, of s ize d ouble-logarithmic in the num b er of vertices. • Distinct regular appro ximations 58 The notio n of f -regularit y , whic h app ears in the regular approximat ion lemma, has since ap- p eared elsewhere. Alon, Shapira, and S ta v [9] inv estigate the q u estion of determin in g if a graph G = ( V , E ) can ha v e distinct regular partitions. Tw o equitable partitions U : U = U 1 ∪ . . . ∪ U k and V : V = V 1 ∪ . . . ∪ V k in to k parts are said to be ǫ -isomorphic if there is a p ermutat ion π : [ k ] → [ k ] suc h that for all but at most ǫ  k 2  pairs 1 ≤ i < j ≤ k , | d ( U i , U j ) − d ( V π ( i ) , V π ( j ) ) | ≤ ǫ . They pr o ve that for s ome f ( k ) = Θ  log 1 / 3 k k 1 / 3  and inﬁ nitely man y k , and for ev ery n > n ( k ), there is a graph on n vertice s with t w o f -regular partitions which are not 1 / 4-isomo rphic. On the other h and, they sho w that if f ( k ) ≤ min(1 /k 2 , ǫ/ 2), then an y t w o equitable partitions of G in to k parts whic h are eac h f -regular must b e ǫ -isomorph ic. • Multicolor and directed regularit y and remov al lemmas The pro of of Szemer ´ ed i’s regularit y lemma h as b een extended to giv e multicolo r (see [27]) and directed (see [7]) extensions. These imp ly multicolo r and directed generalizations of the graph remo v al lemma. As discussed in [18], the new p ro of of the graph r emov al lemma with a log- arithmic tow er heigh t extends with a similar b oun d to these v ersions as w ell. Axeno vic h and Martin [11] recen tly extended the strong regularit y lemma in a similar fashion to giv e multicolo r and directed versions, in order to establish extensions of the induced graph remo v al lemma. Our pro of of the induced graph remov al lemma with a tow er-t yp e b ound similarly ext ends to give m ulticolor and directe d v ersions. • On pro ofs of regularity lemmas As noted b y Go w ers, the constructions in [22] not only sho w that the b ound in Szemer ´ edi’s regularit y lemma is necessarily la rge, b u t that, in s ome sense, the pro of is necessary . An y pro of m ust follo w a long sequence of successiv ely ﬁ ner partitions, eac h exponentiall y larger than the previous one. While this notion is hard to mak e pr ecise, it should b e clear to any one who has studied the p ro of of the r egularit y lemma and the low er b ound construction of Go wers. Theorem 1.1 add s further weigh t to this con viction. F urth er m ore, the pro of of T heorem 1.2 sh o ws that an y pro of of the stron g r egularity lemma requires a long sequence of partitions, eac h of tow er-t yp e larger than the p r evious partition. That is, the iterat ed u se of Szemer´ edi’s regularit y lemma is required in an y proof of th e strong r egularit y lemma. Ac knowledgmen t. W e w ould like to th ank Noga Alo n for helpful commen ts. Note added. After th is p ap er was written w e learned that a v arian t of Theorem 1.2 w as also pro v ed, indep end en tly and sim ultaneously , by Kalya nasund aram and Sh ap ir a. In the situati on of Corollary 1.2, their theorem giv es a low er b ound of wo wzer-t yp e in p log ǫ − 1 for the strong regularit y lemma. References [1] M. Ajtai and E . 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