Random greedy triangle-packing beyond the 7/4 barrier

Random greedy tri angle-pac king b eyond the 7/4 barrier T om Bohman ∗ Alan F rieze † Ey al Lub etzky ‡ Abstract The random greedy algorithm for constructing a large pa r tial Steiner -T riple-System is defined as follows. Begin with a complete graph on n vertices and pro ceed to remov e the edges of triang le s one at a time, where each triangle remov ed is chosen uniformly at random out of all r emaining triangles. This sto chastic pro c ess terminates once it arrives at a tria ngle-free gr aph, and a longstanding op en problem is to estimate the final num b er of edges , or equiv ale nt ly the time it tak es the pro cess to conclude. The in tuition that the edge dis tribution is r oughly uniform at a ll times led to a folklore conjecture that the final num b er of edg e s is n 3 / 2+ o (1) with high pr o bability , wher e as the b e st known upper b ound is n 7 / 4+ o (1) . It is no c o incidence that v ar ious metho ds break precisely at the ex p o nent 7/4 as it corres p o nds to the inher ent barrier where co- de g rees b ecome c omparable to the v ariations in their v alues that ar ose ea rlier in the pr o cess. In this w o rk w e significantly improv e upon the previo us bounds by establishing that w.h.p. the nu mber of edges in the final g raph is at most n 5 / 3+ o (1) . O ur approach relies on a system of martingales used to control k ey graph par ameters, where the crucia l new idea is to har ness the self-corr ecting nature of the pro cess in o rder to control these parameter s well beyond the p oint where their ear ly v aria tio n matches the order of their exp ectation. 1 In tro du ction W e consider t h e random greedy algorithm for triangle-pac king. This stochasti c graph pro cess b egins with the graph G (0), set to b e the complete graph on v ertex set [ n ], then p ro ceeds to rep eatedly remo v e the edges of rand omly c h osen triangles (i.e. copies of K 3 ) fr om the graph. Namely , letting G ( i ) denote the graph that remains after i triangles hav e been remo ved, th e ( i + 1)-th triangle remo ve d is chosen uniformly at rand om from th e set of all triangles in G ( i ). Th e pro cess terminates at a triangle-free graph G ( M ). In th is w ork we study the random v ariable M , the num b er of triangles remo v ed b efore obtaining a triangle-free graph (or equiv alen tly , the num b er of edges in the final triangle-free graph, whic h is  n 2  − 3 M ). This pro cess and its v ariations p la y an imp ortan t r ole in th e history of com binatorics. R¨ odl [13] p ro ve d the Erd˝ os-Hanani conjecture — which p osits the existence of large partial Steiner sy s tems, collect ions of t -sets with the prop ert y that no k -set is a subset of more than one set in the coll ection — in the early 1980’s by w ay of a rand omized construction th at is no w known as the R¨ odl nibb le. This construction is a semi-rand om v ariation on the random greedy pac king pro cess defin ed ab ov e. It is semi-random in the ∗ Department of Ma th ematical S ciences, Carnegie Mellon Universit y , Pittsburgh, P A 15 213. Email add ress: tbohman@ma th.cmu.edu . Research supported in part by NS F gran t D MS-1001638. † Department of Ma th ematical S ciences, Carnegie Mellon Universit y , Pittsburgh, P A 15 213. Email add ress: alan@rando m.math.cmu.edu . Research supp orted in part by NS F gran t D MS-0721878. ‡ Theory Group of Microsoft R esearc h, On e Microso ft W ay , Redmond, W A 98052. Email address: eya l@microsoft.com . 1 sense that the desired ob ject is constructed in a sequence of substan tial pieces, where the pro of of the existence of eac h piece is an application of the probabilistic metho d. Such semi-random constru ctions ha v e b een su ccessfu lly applied to establish v arious ke y r esults in com binatorics (see [1] for an early app lication of this appr oac h and [3] and [12] for fu rther d etails). W e note in passing that semi-random tec hniques ha ve b een refin ed to sh o w the existence of partial Steiner systems that are nearly as large as allo we d b y the simple v olume b ound, see [11] and [17]. In particular, Alon, Kim and Sp encer [2] us ed such tec hn iques to prov e the existence of a set of edge-disjoin t triangles on n v ertices that co v ers all bu t O ( n 3 / 2 log 3 / 2 n ) edges of the complete graph. Despite the success of the R¨ odl nibb le, the limiting b eha vior of the random greedy p ac king pro cess itself r emains u n kno wn , ev en in th e sp ecial case of triangle pac king considered here. Recall that G ( i ) is t h e graph remainin g after i t r iangles ha ve b een remov ed . Le t E ( i ) be the edge set of G ( i ). Note that | E ( i ) | =  n 2  − 3 i and that E ( M ) is the num b er of edges in the triangle-free graph pro d uced b y the pro cess. It is w idely b eliev ed that th e graph pr o duced b y the random greedy triangle-pac king p ro cess b ehav es lik e the Erd˝ os-R ´ enyi random graph with the same edge density , hence the pro cess should end once its num b er of remaining ed ges b ecomes comparable to the n umb er of triangles in the corresp onding Erd˝ os-R ´ enyi rand om graph (i.e., once | E ( M ) | matc hes the order of ( | E ( M ) | /  n 2  ) 3  n 3  ). Throughout th e pap er an ev ent is said to hold with high pr ob ability (w.h.p.) if its probabilit y tends to 1 as n → ∞ . Conjecture (F olklo re) . With high pr ob ability | E ( M ) | = n 3 / 2+ o (1) . Jo el S p encer has offered $200 for a resolution of th is qu estion. It wa s shown by Sp encer [15] in 1995, and ind ep endently by R ¨ odl and T homa [13] in 1996 , that | E ( M ) | = o ( n 2 ) w.h.p. Grable [10] impro v ed this b ound to | E ( M ) | ≤ n 11 / 6+ o (1) via an adaptation of th e R¨ odl nibb le metho d and fu rther sk etc hed ho w similar arguments us in g more d elicate calculations should extend this to a b ound of n 7 / 4+ o (1) w.h.p. W ormald [18] later d emonstrated ho w the differen tial equation m etho d can also giv e nontrivia l b ound s for this problem (as well as generalizations of it), and namely that | E ( M ) | ≤ n 2 − 1 57 + o (1) . Finally , in a companion p ap er [5] that in tro d uced a differen tial equation approac h to this pro cess exploiting its self- correction natur e, the found ations of the present work, the authors ga v e a short pro of that | E ( M ) | = O ( n 7 / 4+ o (1) ) w.h.p. It is imp ortant to note that the p oint at w hic h there are roughly n 7 / 4 remaining edges is a natural barrier in th e analysis of th is pr o cess. T o illustrate this, notice that if the ( i + 1)-st triangle tak en is abc then the c hange in the num b er of triangles in the graph once abc is remo ved is simp ly −| N a,b ( i ) | − | N a,c ( i ) | − | N b,c ( i ) | + 2, wh ere N u,v ( i ) denotes the common neighborho o d of the v ertices u, v ∈ [ n ] in the graph G ( i ). Hence, a n atural prerequisite to analyzing this pro cess is the und erstanding of the co-degrees | N u,v | for all u, v . Supp ose for the sak e of this discussion that early in the ev olution of the pro cess G ( i ) closely resembles the r andom graph with the same n umb er of edges; that is, supp ose G ( i ) is rou gh ly the same as G n,p where p = p ( i ) = 1 − 3 i/  n 2  . If this is th e case when p is close to 1 / 2 (i.e. i is nearly  n 2  / 6) then we exp ect th e | N u,v | ’s to b e close to n/ 4 with v ariations as large as √ n . I f these v ariations in co-degrees p ers ist to the p oint where p = n − 1 / 4 (that is, i roughly  n 2  / 3 − n 7 / 4 ), where w e exp ect the | N u,v | ’s themselves to b e roughly n 1 / 2 , then these v ariations would b e as large as their av erage v alue. Once this happ ens all con trol o ve r co-degrees is lost, e.g. one could ha ve all co-degrees 0 with non-v an ish ing probabilit y , or h alf of the co-degrees 0 and the other half around n 1 / 2 , etc. I n any case, if th e v ariations in | N u,v | that develo p early in th e pro cess are not someho w dealt with, one would exp ect the analysis to br eak do wn once n 7 / 4 edges remain. P erhaps this is the reason th at W ormald [18], who also treated this pr o cess with the differentia l equation metho d , stated that “some n on-trivial mo d ification w ould b e required to equal or b etter Grable’s result.” 2 In th is w ork we exploit the self-correcting n ature of the pro cess in a system of carefully constructed martingales whic h allo ws us to tight en the con trol o ve r k ey graph p rop erties o v er time and ov ercome th e v ariations in their v alues that arise early in the pro cess. Our main resu lt is an u pp er b ound on | E ( M ) | that is significan tly b etter than n 7 / 4 . Theorem 1. Consider the r andom gr e e dy algorithm for triangle-p acking on n vertic e s. L et M b e the numb er of steps it takes the algorithm to terminate and let E ( M ) b e the e dges of the r esulting triangle-fr e e gr aph. Then with high pr ob ability | E ( M ) | = O  n 5 / 3 log 4 n  . A key feature of our pro of of T heorem 1 is an estimate for | N u,v | in which the v ariation de cr e ases as the pr o cess evo lve s. W e stress that estimates for ran d om graph pro cesses with this prop erty are not commonly obtained b y martingale arguments or the differential equation m etho d. The remainder of th e pap er is organized as follo ws. In th e n ext section we d iscuss our analysis of this pro cess in more d etail, listing the random v ariables that w e trac k and the estimates on them th at we are able to p ro ve . The p ro of of our main resu lt, Theorem 2.1, follo ws in Section 3. T heorem 1 follo ws directly from Theorem 2.1. 2 Self-correcting Estimate s Let ( F i ) b e th e filtration give n b y the underlying pro cess. W e note in passing that our probabilit y space is the set of all maximal sequences of edge-disjoint triangles on v ertex set [ n ] with pr obabilit y m easur e giv en b y the uniform random c hoice at eac h step. F or u, v, w ∈ [ n ] d efine N u = N u ( i ) = { x ∈ [ n ] : xu ∈ E ( i ) } , let N u,v = N u ∩ N v and let N u,v ,w = N u ∩ N v ∩ N w . Our main in terest is in trac king t h e num b er of triangles in G ( i ) and the v ariables Y u,v ( i ) = | N u,v ( i ) | = |{ x ∈ [ n ] : xu ∈ E ( i ) , xv ∈ E ( i ) }| . In the course of our argumen t w e will also need to consider the v ariables Y u ( i ) = | N u ( i ) | and Y u,v ,w ( i ) = | N u,v ,w ( i ) | . W e b egin b y writing the one-step exp ected c han ges in our main v ariables of in terest. F or an y rand om v ariable W le t ∆ W be the one-step c hange ∆ W = W ( i + 1) − W ( i ). Let Q ( i ) b e the num b er of triangles in G ( i ). W e hav e E [∆ Y u,v | F i ] = − X x ∈ N u,v Y u,x + Y v,x − 1 { uv ∈ E } Q , (2.1) E [∆ Q | F i ] = − X xy z ∈ Q Y x,y + Y x,z + Y y , z − 2 Q . (2.2) W e use these one-step exp ected changes to relate th e random v ariables to fun ctions of a con tinuous ‘time’ (follo w ing the appr oac h to the differen tial equation metho d introd uced in [4]). W e c ho ose the time-scaling t = t ( i ) = i/n 2 . F ollo wing the conv ention established in the Introduction we set p = p ( i, n ) = 1 − 6 i n 2 = 1 − 6 t . (2.3) 3 Note that p can now b e view ed as either a fu n ction of i or the con tinuous time t ; we pass b et wee n these in terpretations without further comment throughout the pap er. No w, these c hoices yield the tra jectories Y u,v ( i ) ≈ y ( t ) n and Q ( i ) ≈ q ( t ) n where we set y ( t ) = p 2 ( t ) and q ( t ) = p 3 ( t ) / 6 . W e can arrive at these equatio n s by either d eriving them from (2.1) and (2.2) and the assump tion th at the one-step c hanges in the tra jectory are equal to the exp ected one-step c hange in the corresp onding random v ariable or by app ealing to our G n,p in tuition. The companion pap er [5] uses these v ariables alone to establish a b oun d of O ( n 7 / 4 log 5 / 4 n ) on the num b er of edges that surviv e to the conclusion of the algorithm. In order to ac hiev e b etter precision, w e in tro d uce additional v ariables with the cent r al goal of estab- lishing an estimate for Y u,v with v ariation that decreases as the pro cess ev olv es. (F or applications of the differen tial equation metho d that exploit this kind of ‘self-correcting’ phenomenon, see [7] and [16].) W e w ould like to add r andom v ariables to our collection that will giv e us b etter con trol on the expr ession in the numerator of (2.1), the one-step exp ected c h ange in Y u,v . T o this end we tak e a closer look at this expression. W e hav e X x ∈ N u,v ( Y u,x + Y v,x − 1 { uv ∈ E } ) = R u,v + R v,u + Y u,v 1 { uv ∈ E } (2.4) where R u,v =    { ( x, y ) : xy ∈ E , x ∈ N u,v , y ∈ N u and y 6 = v }    . (Notice that R u,v coun ts ordered pairs, thus edges in N u,v are counted twic e.) W e exp ect to h a v e R u,v ≈ p Y u Y u,v whic h in turn suggests th at P x ∈ N u,v ( Y u,x + Y v,x − 1 { uv ∈ E } ) ≈ p Y u,v ( Y u + Y v ). Th is expression is in a form that should provide self-correction in our estimate f or Y u,v . Ind eed, if Y u,v is large compared to its av erage then s o will this term b e and so (as this term is n egated in the exp ected on e-step c hange) Y u,v will h a v e a drift bac k tow ard its mean. This approximati on emph asizes th e n eed to con trol v ertex degrees: turning to Y u w e hav e E [∆ Y u | F i ] = − 1 Q X x ∈ Y u Y u,x = − 2 T u Q where T u =    { xy ∈ E ( i ) : x, y ∈ N u }    . The v ariable T u again lets us bypass th e accumulatio n of worst case individual errors in the s u mma- tion of Y u,v v ariables. W e exp ect to h a v e T u ≈ p  Y u 2  . Finally , cont rol o ver trip le-degrees Y u,v ,w is further n eeded for our concen tration argumen ts to hold. W e th us arrive at t h e ensem ble of v ariables Q, Y u,v , R u,v , Y u , T u and Y u,v ,w for all u, v , w ∈ V G . The follo wing th eorem establishes concentrat ion for this ensem b le (throughout the pap er A = B ± C is short f or A ∈ [ B − C , B + C ]). Theorem 2.1. Set γ = 1 2 , ˆ γ = ˆ γ ( n ) = γ − 6 log n and Φ = Φ ( p, n ) = e 1 − p log 2 n . Then ther e exist absolute c onstants α, β , κ, µ > 0 such that with high pr ob ability Q = n 3 p 3 / 6 ± α 2 n 2 p 2ˆ γ − 1 Φ 2 (2.5) Y u,v = np 2 ± αn 1 / 2 p ˆ γ Φ (2.6) 4 R u,v = pY u Y u,v ± β n 3 / 2 p 2+ˆ γ Φ (2.7) Y u = np ± κn 1 / 2 p ˆ γ − 1 Φ (2.8) T u = pY 2 u / 2 ± µn 3 / 2 p 1+ˆ γ Φ (2.9) Y u,v ,w = np 3 ± 2 q np 3 log 5 n (2.10) for al l u, v , w and as long as p ≥ p ⋆ =  6 α 2 e 2 log 10 n n  1 / (4 − 2ˆ γ ) . T o d educe T heorem 1, observe that p ⋆ , defined as the smallest p for which the theorem h olds, satisfies p ⋆ = O  n − 1 / 3 log 10 / 3 n  (2.11) since 4 − 2 ˆ γ = 3 + O (1 / log n ). In p articular, p = p ⋆ satisfies n 3 p 3 / 6 > α 2 n 2 p 2ˆ γ − 1 Φ 2 since we hav e Φ = e (1 − p ) log 2 n ≤ e log 2 n . It thus follo ws that Q > 0 w.h.p. due to (2.5) and it remains to reco ve r th e n umb er of ed ges corresp onding to p ⋆ . Recalling (2.3) w e hav e | E ( i ) | =  n 2  − 3 i =  n 2  − 1 2 (1 − p ⋆ ) n 2 = 1 2  n 2 p ⋆ − n  and the desired result follo ws from (2.11) with ro om to spare in the p o wer of the logarithmic factor. W e pr o v e Th eorem 2.1 in the follo wing section b y app lying martingale argumen ts to rand om v ariables that trac k the differences b et wee n the random v ariables w e are in terested in and the v ariables they should follo w. Note that we establish some form of self-correction for eve r y v ariable in this ensemble, with the notable exception of Y u . The authors susp ect that the metho d s in tro d uced in this pap er can b e f urther dev elop ed to achiev e b etter high p r obabilit y upp er b oun ds on | E ( M ) | . T h is might b e ac h iev ed by expandin g the en sem ble of random v ariables (p erhaps usin g ensem bles of generalized extension-count ing v ariables, wh ic h is the approac h tak en in the recen t a n alysis of the H -fr ee pro cess [6]). Ho w eve r, it see ms that a non trivial mo dification wo uld b e needed to p ro ve the conjectured b ound | E ( M ) | = n 3 / 2+ o (1) . F or notational con v enience we set Λ = 1 log 2 n . Note th at wh ile Thereom 2.1 app lies, estimates (2.6)–(2.9) an d (2.5) can eac h b e written as a main term times (1 + O (1 / log 2 n )) = (1 + O (Λ)). Throughout the p ap er we will u s e a con ven tion whereby all Gr eek letters are universal constan ts. W e d o not replace an y of the constant s (includ ing the p iv otal γ ) with their actual v alues. T his is done in the interest of und er s tanding the role these constan ts p la y in the calculatio ns ; it tu r ns out that these constan ts m ust b e balanced in a fairly d elicate w a y . W e observ e that these constan ts can tak e th e actual v alues α = 1 , β = 1 2 , δ = 1 3 , κ = 1 4 , µ = 1 4 . The k ey conditions these constan ts m us t s atisfy are (3.17), (3.19) , (3.23) and (3.24). 3 Pro of of Theorem 2.1 Define p ⋆ as in (2.11) and let i ⋆ = 1 6 (1 − p ⋆ ) n 2 b e the analogous roun d. Let G i b e the ev ent that all estimates in Theorem 2.1 hold for th e fi rst i s teps of the pro cess. 5 F or e ac h v ariable and eac h b ound (upp er and lo wer) in Theorem 2.1 we defin e a critic al interval . This interv al has one end at the b oun d w e are tryin g to maint ain and the other end sligh tly closer to the exp ected tr a jectory of the r andom v ariable. If one of the estimat es of T heorem 2.1 is violated then the corresp onding random v ariable ‘crosses’ a critical interv al. W e b oun d the pr obabilit y of eac h su c h ev en t usin g a martingale argu m en t, introd ucing a separate sup erm artingale f or eac h v ariable and b ound of interest and for eac h step in wh ic h the v ariable could enter the critical in terv al. Theorem 2.1 then follo ws fr om the union b ound (note that the num b er of sup ermartingales w e consider is b oun d ed by a p olynomial in n ). W e restrict our attent ion to these critical interv als b ecause the exp ected one-step c hanges in our rand om v ariables eac h ha v e a ‘drift’ term that push es a w a yward v ariable bac k to ward th e exp ected tra jectory . By only considering the critical interv als w e mak e full use of these terms : T his is the mec hanism we are u sing to establish self-correcting estimate s. F or an application of this idea in a setting with fewe r v ariables, see [5]. Let the stopping time τ b e the minim um of i ⋆ and the smallest index i suc h that G i do es n ot hold. Consider an ev ent E of th e form X ( i ) ≤ x ( t ) for all i ≤ i ⋆ where w e assum e that X ( i ) is a rand om v ariable and x ( t ) is not. Note that ev ery b ound in (2.5)–(2.10) can b e w ritten in this form; that is, the eve nt { τ = i ⋆ } can b e written { τ = i ⋆ } = \ ℓ ∈I E ℓ where |I | is p olynomial in n and eac h ev ent E ℓ is of the form X ( i ) ≤ x ( t ) for all i ≤ i ⋆ . F or ea ch su c h ev en t E we int ro duce a critical in terv al of the form I E ( t ) = ( x ( t ) − w ( t ) , x ( t )) wh ere w ( t ) = o ( x ( t )). Consider a fixe d step i 0 , whic h we view as a s tep at which X ( i ) might en ter the critical interv al I E . Set t 0 = i 0 /n 2 . Define th e stopping time τ E ,i 0 to b e the minimum of m ax { i 0 , τ } and the smallest i ≥ i 0 suc h that X ( i ) 6∈ I E . Note that if X ( i 0 ) 6∈ I E ( t 0 ) then we ha ve τ E ,i 0 = i 0 . Thus this stopp ing time is (formally) w ell-defined on the fu ll probability space (n.b. we only make use of this stopping time when X ( i 0 ) is in the critical int erv al and X ( i 0 − 1) is not). W e no w establish a b ound B ( i ) on the one-step c hange in X ( i ) conditioned on G i . This b ound is far less than the width w ( t ) of the critical in terv al. Give n a p articular ev en t E and starting step i 0 , we w ork with the sequence of random v ariables Z E ,i 0 ( i ) = Z ( i ) = ( X ( i ) − x ( t ) if i 0 ≤ i ≤ τ E ,i 0 Z ( i − 1) otherwise. Note that if X ( i 0 − 1) is not in the critical int erv al (and i 0 < τ ) th en Z ( i 0 ) < − w ( t 0 ) + B ( i 0 ). Therefore, in the ev en t E c there are s teps i 0 < j ≤ i ⋆ suc h that Z ( i 0 ) < − w ( t 0 ) + B ( i 0 ) and Z ( j ) ≥ 0. Ho wev er, our stopping time τ stops all of these sequ ences as so on as an y of our conditions (2.5)–(2.10) are violated. So, w e hav e { τ < i ⋆ } ⊆ [ ℓ ∈I [ 1 ≤ i 0 w ( t 0 ) − B ( i 0 ) } = [ ℓ ∈I [ 1 ≤ i 0 w ( t 0 ) − B ( i 0 ) } . It remains to b ound th e p robabilit y of eac h ev ent in this union. This is done for eac h of the b ounds (2.5)–(2.10) in tur n in Sections 3.1 – 3.6. In order to b ound the probability of these eve nts w e will apply the follo wing inequalit y d ue to F reedman [9], whic h was originally stated for m artingales yet its pr o of extends essen tially u nmo difi ed to sup ermartingales. 6 Theorem 3.1 ([9], Thm 1.6) . L et ( S 0 , S 1 , . . . ) b e a su p ermartingale w.r.t. a filter ( F i ) . Supp ose that S i +1 − S i ≤ B for al l i , and write V t = P t − 1 i =0 E  ( S i +1 − S i ) 2 | F i  . Then for any s, v > 0 P  { S t ≥ S 0 + s, V t ≤ v } for some t  ≤ exp  − s 2 2( v + B s )  . Our applications of this inequalit y will eac h ha ve tw o parts: a careful calculation that establishes a martingale condition and a coarser argu m en t th at p ro vides b ounds on b oth the one-step change s and the second momen t of the one-ste p c hanges of these v ariables. W e emphasize that our carefully c hosen stopping times allo w us to assume that th e eve nt G i holds thr oughout th ese calculations. This is h enceforth assumed without further commen t. 3.1 Edges b etw een a co-neigh b or ho o d and a neigh b orho o d ( R u,v ) W e b egin with an analysis of th e one-step exp ected c hange. There are 7 t yp es of triangles that con tribute to E [∆ R u,v | F i ]. (1) T riangles v xy wh ere x ∈ N u,v and y ∈ N v \ N u,v and y 6 = u . F or x ∈ N u,v there are Y v,x − Y u,v ,x − 1 { uv ∈ E } suc h triangles and selection of one of these triangles mo ves x from N u,v to N u \ N u,v and thereby decreases R u,v b y Y u,x − 1 { uv ∈ E } . T his r esults in a contribution of − X x ∈ N u,v Y v,x − Y u,v ,x − 1 { uv ∈ E } Q  Y u,x − 1 { uv ∈ E }  = − X x ∈ N u,v Y v,x − Y u,v ,x Q Y u,x + O  p n  , (3.1) where in the last equalit y w e a bs orb ed the indicator v ariables i nto an O ( p/n ) t erm based on th e estimates of Theorem 2.1 u p to this p oin t. (2) T riangles uxy where x ∈ N u,v and y ∈ N u \ N u,v with y 6 = v . (There are are Y u,x − Y u,v ,x − 1 { uv ∈ E } suc h v ertices y for eac h x ∈ N u,v and there are Y u,v ,y suc h ve rtices x for eac h y ∈ N u \ ( N u,v ∪ { v } ).) The selection of suc h a triangle remo v es y fr om N u while moving x from N u,v to N v \ N u,v . The effect of y is a decrease of Y u,v ,y in R u,v while the effect of x is an additional d ecrease of Y u,x + Y u,v ,x − 1 { uv ∈ E } − 1 (the v ertex y ∈ N u,x w as already counte d ). Th e o v erall con tribution is − X x ∈ N u,v Y u,x − Y u,v ,x Q ( Y u,x + Y u,v ,x ) − X y ∈ N u \ N u,v Y 2 u,v ,y Q + O  p n  , (3.2) where the indicator terms w ere again absorb ed in to th e O ( p/n ) term. (3) T riangles v xy where x, y ∈ N u,v . Cho osing suc h a triangle mo ve s x , y fr om N u,v to N u \ N u,v and thus decreases R u,v b y ( Y u,x − 1 { uv ∈ E } ) + ( Y u,y − 1 { uv ∈ E } ). Th is contributes − X x,y ∈ N u,v xy ∈ E 1 Q ( Y u,x + Y u,y ) + O  p 2 n  = − X x ∈ N u,v Y u,v ,x Q Y u,x + O  p 2 n  . (3.3) (4) T riangles uxy where x, y ∈ N u,v . Selectio n of such a triangle mo ves x, y f rom N u,v to N v \ N u,v and so decreases R u,v b y ( Y u,x + Y u,v ,x − 1 { uv ∈ E } ) + ( Y u,y + Y u,v ,y − 1 { uv ∈ E } ) − 2, translating to − X x,y ∈ N u,v xy ∈ E 1 Q ( Y u,x + Y u,v ,x + Y u,y + Y u,v ,y ) + O  p 2 n  = − X x ∈ N u,v Y u,v ,x Q ( Y u,x + Y u,v ,x ) + O  p 2 n  . (3.4) 7 (5) T riangles uxy where xy is in the set of edges ind u ced by N u \ N u,v . Eac h s u c h triangle d ecreases R u,v b y Y u,v ,x + Y u,v ,y . T his r esults in a contribution of − X x,y ∈ N u \ N u,v xy ∈ E 1 Q ( Y u,v ,x + Y u,v ,y ) = − X x ∈ N u \ N u,v Y u,x − Y u,v ,x Q Y u,v ,x . (3.5) (6) T riangles uv x for x ∈ N u,v con tribute − 1 { uv ∈ E } X x ∈ N u,v Y u,x + Y u,v ,x − 1 Q = − R u,v 1 { uv ∈ E } Q = O  p n  . (3.6) (7) T riangles xy w where x ∈ N u,v and y ∈ N u while w 6 = u . W e note in passing that th is is the only typ e of triangle whose selection impacts R u,v while changing neither Y u − Y u,v nor Y u,v . F or a fixed xy ∈ E with x ∈ N u,v and y ∈ N u there are Y x,y − 1 such triangles, and eac h w ould decrease R u,v b y either 1 or 2. Merely applying our b ound s on eac h Y x,y term ind ividually wo u ld pro d uce an un d esirable error. Instead, w e will su m ov er x ∈ N u,v and use our error b ounds on R x,u . This should giv e a b etter estimate by aggregating multiple Y x,y terms for b etter cum ulativ e error b ound s, and indeed this prov es to b e a crucial c hoice. A triangle xy w in this category will decrease R u,v b y 1 if uw / ∈ E and b y 2 if uw ∈ E . Recall that R x,u coun ts the n umber of edges b et w een N x \ N u and N x,u plus t wice the num b er of edges within N x,u . Hence, the contribution in this case is precisely ( − 1 /Q ) times  X x ∈ N u,v R x,u  − R v,u 1 { uv ∈ E } = − X x ∈ N u,v h pY x Y u,x ± β n 3 / 2 p 2+ˆ γ Φ i + O  n 2 p 4  = X x ∈ N u,v [ pY x Y u,x ] ± (1 + O (Λ)) β n 5 / 2 p 4+ˆ γ Φ , where in the last equalit y we absorb ed the O ( n 2 p 4 ) error term using the fact that Λ = log − 2 n is O ( n 1 / 2 p ˆ γ ) (with room to spare). Plugging in the fac t that Y x = np ± κn 1 / 2 p ˆ γ − 1 Φ and using the iden tit y R u,v = P x ∈ N u,v ( Y u,x − 1 { uv ∈ E } ), w e can conclude that the con tribution in this case is − R u,v np 2 Q ± 1 Q h (1 + O (Λ)) κn 5 / 2 p 4+ˆ γ Φ + (1 + O (Λ)) β n 5 / 2 p 4+ˆ γ Φ i , (3.7) where w e absorb ed all ind icator v ariables (at most np 2 Q Y u,v = O ( p/n )) in to the fin al err or term. No w that we h a v e an expression (alb eit in 7 parts) for the exp ected c hange in R u,v , w e are r eady to consider the exp ected c hange in R u,v relativ e to its exp ected tra j ectory . Define X = R u,v − pY u Y u,v and consider E [∆ X | F i ]. W r ite ∆ [ pY u Y u,v ] = p ( i + 1)∆ [ Y u Y u,v ] + ∆ pY u ( i ) Y u,v ( i ) . (3.8) W e w ill see that the exp ected change in R u,v due to triangles of typ es 1–6 will balance off with the first term in (3.8), while the exp ected c hange du e to triangles of t yp e 7 will b e balanced b y the second term. Collecting (3.1)–(3.6), the total con trib u tion to E [∆ R u,v | F i ] f rom triangles of t yp es 1–6 equals − 1 Q X x ∈ N u,v Y u,x ( Y u,x + Y v,x ) − 1 Q X x ∈ N u Y u,x Y u,v ,x + O  p n  . (3.9) F urthermore, we can analyze the c h an ge in Ξ = Y u,v Y u b y consider in g th e follo wing 3 cases: 8 (i) Selecti n g a triangle uxy f or x, y ∈ N u \ N u,v : h ere ∆ Y u,v = 0 wh ile ∆ Y u = − 2 an d so ∆Ξ = − 2 Y u,v . (ii) Selecting a triangle v xy for x ∈ N u,v and y 6 = u : the co-neighborho o d loses x and in addition loses y if y ∈ N u,v , while Y u remains u nc hanged. Th us ∆ Ξ = − 2 Y u if y ∈ N u,v and ∆ Ξ = − Y u otherwise. (iii) Selecting a triangle uxy for x ∈ N u,v : If y ∈ N u,v then ∆ Y u,v = ∆ Y u = − 2 and ∆Ξ = − 2 Y u − 2 Y u,v + 4. Similarly , if y = v then ∆Ξ = − Y u − Y u,v + 1. Otherwise, y ∈ N u \ N u,v and ∆ Y u,v = − 1 while ∆ Y u = − 2, hence ∆Ξ = − Y u − 2 Y u,v + 2. Altoget h er, we can obtain the factors of 2 in Item (i) and in the case y ∈ N u,v of Items (ii),(iii) automat- ically b y sym metry when s u mming ov er x as follo ws: E  ∆[ Y u,v Y u ] | F i  = − Y u,v X x ∈ N u \ N u,v Y u,x − Y u,v ,x Q − Y u X x ∈ N u,v Y v,x − 1 { uv ∈ E } Q −  ( Y u + Y u,v + O (1))  X x ∈ N u,v Y u,x Q  +  ( Y u + 2 Y u,v ) − ( Y u + Y u,v ) + O (1)   X y ∈ N u \ N u,v Y u,v ,y Q   . All the triple-degree terms cancel out and we can colle ct all the O (1)-terms and rewrite the ab o ve as − 1 Q X x ∈ N u,v Y u ( Y u,x + Y v,x ) − 1 Q X x ∈ N u Y u,v Y u,x + O  1 n  . Notice that when multiplying the ab ov e by p the err or term b ecomes an add itiv e O ( p/n ) while the main terms are of order O ( p 2 ). As such, the same estima te holds for the result of multiplying the ab o v e b y p ( i + 1) (wh ic h d iffers from p ( i ) b y an additive O ( n − 2 ) error and th us introd uces an extra O (( p/n ) 2 ) error term). W e can no w com bin e this with the c h ange in R u,v giv en in (3.9 ) to get that th e con tribu tion to E [∆ X | F i ] f rom triangles of t yp es 1–6 and the first term in (3.8) is − 1 Q X x ∈ N u,v ( Y u,x + Y v,x ) ( Y u,x − pY u ) − 1 Q X x ∈ N u Y u,x ( Y u,v ,x − pY u,v ) + O  p n  . (3.10) In order to rewrite the last tw o summations, w e need the follo wing straigh tforwa r d estimate: Lemma 3.2. L et ( x i ) i ∈ I and ( y i ) i ∈ I such that | x i − x j | ≤ δ 1 and | y i − y j | < δ 2 for al l i, j ∈ I . Then     X i ∈ I x i y i − 1 | I |  X i ∈ I x i  X i ∈ I y i      ≤ | I | δ 1 δ 2 . The ke y observ ation here is that the first an d second su mmations in (3.10) feature the random v ariable X itself, a fact wh ic h our self-correction argumen t for X hinges on. Namely , by definition of R u,v w e ha ve P x ∈ N u,v ( Y u,x − 1 { uv ∈ E } − pY u ) = R u,v − pY u Y u,v = X and similarly P x ∈ N u ( Y u,v ,x − pY u,v ) = X . By the lemma ab o ve an d our err or estimates from Th eorem 2.1, the firs t summ ation in (3.1 0 ) is equal to − ( R u,v + R v,u ) X Q Y u,v + O  n − 1 p 2ˆ γ − 1 Φ 2  = − 12 + O (Λ) n 2 p X + O  Φ 2 n  , where th e last equalit y used our (1 + O (Λ))-appro ximation for ( R u,x + R v,x ), Q and Y u,v . Similarly , the second summation in (3.10) can b e estimated by − 2 T u X Q Y u + O  np Q · n 1 / 2 p ˆ γ Φ · q np 3 log 5 n  = − 6 + O (Λ) n 2 p X + O Φ log 5 / 2 n n ! . 9 Altoget h er, the con tribution to E [∆ X | F i ] f rom triangles of t yp es 1–6 and the first term in (3.8) is − 18 + O (Λ) n 2 p X + O  log 5 n n  . (3.11) W e now turn to th e triangles of t yp e 7. As we noted ab o ve , w e b alance the term (3.7) with the second term in (3.8 ), i.e. th e exp ected change in pY u Y u,v due to the c hange in p (which deterministically decreases b y 6 /n 2 ). Th e total contribution to E [∆ X | F i ] from these terms is − R u,v np 2 Q + 6 n 2 Y u Y u,v ± 1 Q (1 + O (Λ))( β + κ ) n 5 / 2 p 4+ˆ γ Φ = − 6 + O (Λ) n 2 p X ± (6 + O (Λ))( β + κ ) p 1+ˆ γ Φ n 1 / 2 . (3.12) The com bination of (3.11),(3.12) giv es E [∆ X | F i ] = − 24 + O (Λ) n 2 p X ± (6 + O (Λ))( β + κ ) p 1+ˆ γ Φ n 1 / 2 , (3.13) where the O (Λ) n − 1 / 2 p 1+ˆ γ Φ term absorb ed the error-term in (3.11) b y the c hoice of p in (2.11). W e are no w ready to establish the concentrat ion of R u,v via a martingale argumen t. As outlined ab o ve, we in tro duce t wo critical int erv als for the r andom v ariable X , corresp onding to the u p p er b ound and low er b ound on R u,v . These in terv als hav e one endp oint at the b ound we are trying to establish and the other somewhat closer to zero (corresp onding to th e exp ected tra jectory of X ). F o r the v ariable R u,v to violate Eq. (2.7) it must b e th at X crosses one of the critical in terv als. Our critical in terv al for the u pp er b ound on R u,v is I R =  ˆ β n 3 / 2 p 2+ˆ γ Φ , β n 3 / 2 p 2+ˆ γ Φ  where ˆ β =  1 − log − 1 n  β . (3.14) Supp ose th at X ( i 0 ) ente r s I R for the first time at round i 0 (within the time range cov ered by Theorem 2.1) and define the stopping time τ R = min { i ≥ i 0 : X ( i ) < ˆ β n 3 / 2 p 2+ˆ γ Φ } , i.e. th e first time b eyond i 0 at which X exits the inte r v al through its lo wer endp oint. W e claim that Z ( i ∧ τ R ) is a s up er m artingale, w h ere Z ( i ) = X ( i ) − β n 3 / 2 p 2+ˆ γ Φ for i ≥ i 0 . T o see this, write t = i/n 2 according to w hic h p = 1 − 6 t and Φ = e 6 t log 2 n , and note that for an y ˆ γ > 0 the second deriv ativ e of f ( t ) = e 6 t (1 − 6 t ) 2+ˆ γ is uniformly b ound ed in [0 , 1 6 ], hence f  t + n − 2  = f ( t ) +  6 e 6 t (1 − 6 t ) 2+ˆ γ − 6(2 + ˆ γ ) e 6 t (1 − 6 t ) 1+ˆ γ  n − 2 + O ( n − 4 ) . (3.15) This pr ovides an estimate for ∆[ β n 3 / 2 p 2+ˆ γ Φ] b et we en Z ( i + 1) and Z ( i ). At the same time, for any X ( i ) satisfying (3.14 ) we can plug in the low er b ound this giv es for X in (3.13) and obtain E [∆ Z | F i , τ R > i ] ≤ −  24 + O (Λ)  ˆ β p 1+ˆ γ Φ n 1 / 2 +  6 + O (Λ)  ( β + κ ) p 1+ˆ γ Φ n 1 / 2 +  1 + O ( n − 2 )   6(2 + ˆ γ ) β p 1+ˆ γ Φ n 1 / 2 − 6 β p 2+ˆ γ Φ n 1 / 2  =  − 4 ˆ β + 3 β + κ + ˆ γ β + O (Λ)  6 p 1+ˆ γ Φ n 1 / 2 +  − 6 β + O (Λ)  p 2+ˆ γ Φ n 1 / 2 . 10 Since ˆ β = (1 − log − 1 n ) β and ˆ γ = γ − 6 log − 1 n w e hav e − 4 ˆ β + 3 β + κ + ˆ γ β + O (Λ) = − (1 − γ ) β + κ − 2 β log n + O (Λ) < − (1 − γ ) β + κ − β log n , where in the inequalit y absorb ed the O (Λ) term int o a s in gle ( β / log n )-term for large enough n . Alto- gether, we conclude that if n is large en ough then Z ( i ∧ τ R ) is a s up er m artingale provided that (1 − γ ) β + β log n ≥ κ (3.16) and Λ = o ( β ). I n p articular, w e can relax this condition in to the requirement that Λ = o ( β ) and (1 − γ ) β ≥ κ . (3.17) T o apply F reedman’s inequalit y w e need to obtain b ounds on | ∆ Z | and E [(∆ Z ) 2 | F i ]. Recall that ∆ Z = ∆ R u,v − ∆[ p Ξ ] − ∆[ β n 3 / 2 p 2+ˆ γ Φ] where Ξ = Y u Y u,v . By (3.8) and the fact that ∆ p = − 6 /n 2 w e ha v e ∆[ p Ξ] = p ∆Ξ + O ( p 3 ), w h ile ∆ [ β n 3 / 2 p 2+ˆ γ Φ] = O ( n − 1 / 2 p 1+ˆ γ Φ) = o (1). Hence, ∆ Z will b e d ominated b y ∆ R u,v − ∆[ p Ξ]. There are four cases to consider here: (i) Cho osing a triangle that includes u or v and some v ertex x ∈ N u,v (triangles of typ es 1–4,6 in the analysis of E [∆ R u,v | F i ] ab ov e): There are O ( n 2 p 4 ) such triangles and selecting one of them affects b oth R u,v and Ξ. As we next sp ecify , the principle terms in these c hanges are id en tical and so | ∆ Z | is b oun d ed by the error terms in our approximat ions. Ind eed, going bac k to th e analysis of the triangle t yp es as w ell that of ∆Ξ we recall the v arious triangle t yp es satisfied: • T yp e 1: ∆ R u,v = − Y u,x + O (1) vs. ∆Ξ = − Y u . • T yp e 2: ∆ R u,v = − ( Y u,x + Y u,v ,x + Y u,v ,y ) + O (1) vs. ∆Ξ = − Y u − 2 Y u,v + O (1). • T yp e 3: ∆ R u,v = − ( Y u,x + Y u,y ) + O (1) vs. ∆Ξ = − 2 Y u . • T yp e 4: ∆ R u,v = − ( Y u,x + Y u,v ,x + Y u,y + Y u,v ,y ) + O (1) vs. ∆Ξ = − 2 Y u − 2 Y u,v + O (1). • T yp e 6: ∆ R u,v = − Y u,x − Y u,v ,x + O (1) vs. ∆Ξ = − Y u − Y u,v + O (1). In all of the ab ov e cases the main terms of ∆ R u,v cancel those of p ∆Ξ at the cost of an O ( n 1 / 2 p ˆ γ log 2 n )- error due to the app ro ximations of Y u,v and Y u (dominating all other errors). (ii) Cho osing a triangle that includ es u but no v ertex i n N u,v (triangles of t yp e 5 ab o ve): there are O ( n 2 p 3 ) such triangles and eac h corresp onds to ∆ R u,v = − ( Y u,v ,x + Y u,v ,y ) vs. ∆ Ξ = − 2 Y u,v . Hence, in this case | ∆ Z | = O ( p np 3 log 5 n ) dominated b y triple co-degrees (recalling that γ = 1 2 ). (iii) Cho osing triangles that affect the v alue of R u,v but con tain neither u nor v (t yp e 7 triangles): Eac h of these O ( n 3 p 6 ) triangles corresp onds to | ∆ Z | = O (1) as ∆ R u,v = − 1 and ∆Ξ = 0. (iv) Cho osing an y other triangle: as R u,v and Ξ are b oth unc h anged, these triangles can mo d ify Z by at most O ( p 3 ) due to the add itiv e 6 /n 2 c hange in p . The L ∞ b ound on ∆ Z is clearly dominated by round s of th e fi rst sort an d | ∆ Z | = O ( n 1 / 2 p ˆ γ log 2 n ). F o r an L 2 b ound n otice that the 4 roun d t yp es contribute O ( p 1+2ˆ γ log 4 n ), O ( p 3 log 5 n ), O ( p 3 ) and O ( p 6 ) resp ectiv ely to E [(∆ Z ) 2 | F i ]. As ˆ γ < 1 we ha ve p 3 log 5 n = o ( p 1+2ˆ γ log 5 n ) while the fact that p ≥ p ⋆ (whic h has order n − 1 / (4 − 2ˆ γ )+ o (1) as was seen in Eq. (2.11)) implies that log 5 n n = o ( p 1+2ˆ γ ) since np 1+2ˆ γ ≥ n (3 − 4 γ ) / (4 − 2 γ ) − o (1) > n 1 / 5 for large enough n . Altoget her it f ollo ws that E [(∆ Z ) 2 | F i ] = O ( p 1+2ˆ γ log 4 n ). Clearly the L ∞ and the L 2 b ound s on ∆ Z also hold in the conditional space given τ R > i . Recall that we are intereste d in Z ( i ) starting at time i 0 , i.e. immediately after X en ters the critical in terv al I R . Let p 0 = p ( i 0 ) = 1 − 6 i 0 /n 2 and observ e that our b ound on | ∆ Z | guaran tees t h at 0 ≤ X ( i 0 ) − ˆ β n 3 / 2 ( p 0 ) 2ˆ γ Φ ≤ O ( n 1 / 2 ( p 0 ) ˆ γ log 2 n ). H ence, Z ( i 0 ) ≤ ( ˆ β − β ) n 3 / 2 ( p 0 ) 2+ˆ γ Φ + O ( n 1 / 2 ( p 0 ) ˆ γ log 2 n ) ≤ − 1 2 β n 3 / 2 ( p 0 ) 2+ˆ γ log n , 11 where the final f actor of 1 2 readily cancels the O ( n 1 / 2 ( p 0 ) ˆ γ log 2 n )-term for sufficientl y large n since n ( p 0 ) 2 log − 1 n ≥ n 1 − o (1) ( p ⋆ ) 2 > n 1 / 4 . W e no w app ly F reedman’s inequalit y (Theorem 3 .1) to the su- p ermartingale S j = Z  ( i 0 + j ) ∧ τ R  while noting that the ab o v e analysis implies that S 0 ≤ − 1 2 β n 3 / 2 ( p 0 ) 2+ˆ γ log n , max j | S j +1 − S j | = O  n 1 / 2 ( p 0 ) ˆ γ log 2 n  , X j E  ( S j +1 − S j ) 2 | F ′ j  = O  log 5 n X i ≥ i 0 ( p ( i )) 1+2ˆ γ  = O  n 2 ( p 0 ) 2+2ˆ γ log 5 n  , where F ′ j = F i 0 + j . W e d educe that for s ome fi xed constan t c > 0, P  ∪ j  S j ≥ 0  ≤ exp  − c n 3 ( p 0 ) 4+2ˆ γ log 2 n n 2 ( p 0 ) 2+2ˆ γ log 5 n + n 2 ( p 0 ) 2+2ˆ γ log 3 n  = exp  − cn ( p 0 ) 2 log − 3 n  , whic h is su fficien tly small to afford a u n ion b ound o v er all u, v and time steps i 0 , hence w.h.p. X nev er crosses the critical interv al I R and so X ( i ) ≤ β n 3 / 2 p 2+ˆ γ Φ f or all u, v an d i . The same argument sho ws that w.h.p. X ( i ) ≥ − β n 3 / 2 p 2+ˆ γ Φ for all u, v , i b y considering the low er in terv al  − β n 3 / 2 p 2+ˆ γ Φ , − ˆ β n 3 / 2 p 2+ˆ γ Φ  and analyzing the v ariable Z ( i ) = − X ( i ) − β n 3 / 2 p 2+ˆ γ Φ. Th is completes the p ro of of Eq. (2.7 ). 3.2 Co-degrees ( Y u,v ) F ollo wing the lines of § 3.1 w e will establish concen tration f or X = Y u,v − np 2 . As p decreases b y 6 /n 2 with eac h step, Eq. (2.1) and (2.4) sho w that E [∆ X | F i ] = − X x ∈ N u,v Y u,x + Y v,x − 1 { uv ∈ E } Q + 6 n  2 p − 6 /n 2  = − R u,v + R v,u Q + 12 p n + O  1 n 2 p  , where th e last error term absorb ed the in d icators and O ( n − 3 ) from the first expression. Substituting our estimates (2.7) for R u,v , R v,u w e get that this is equal to − 1 Q  p ( Y u + Y v ) Y u,v ± 2 β n 3 / 2 p 2+ˆ γ Φ  + 12 p n + O  1 n 2 p  , and using the estimate (2.8) for Y u and that Q = (1 + O (Λ)) n 3 p 3 w e can conclude that E [∆ X | F i ] = − p (2 np ± 2 κn 1 / 2 p ˆ γ − 1 Φ) Y u,v Q + 12 p n ± (12 + O (Λ)) β p ˆ γ − 1 Φ n 3 / 2 + O  1 n 2 p  . Crucially , we d id not approximat e Q in the first expression with a (1 + O (Λ)) correction facto r as this w ould incur an error that w ould b e too large to handle. Instead, there we apply Eq. (2.5) and the fact that Y u,v = X + np 2 to obtain that p (2 np ± 2 κn 1 / 2 p ˆ γ − 1 Φ) Y u,v Q = 12 + O (Λ) n 2 p X + 2 n 2 p 4 1 6 n 3 p 3 ± α 2 n 2 p 2ˆ γ − 1 Φ 2 ± (12 + O (Λ)) κp ˆ γ − 1 Φ n 3 / 2 = 12 + O (Λ) n 2 p X + 12 p n  1 + O  p 2ˆ γ − 4 Φ 2 n  ± (12 + O (Λ)) κp ˆ γ − 1 Φ n 3 / 2 . 12 Com binin g this with the ab o v e estimate for E [∆ X | F i ], th e term 12 p/n v anishes and w e get that E [∆ X | F i ] = − 12 + O (Λ) n 2 p X ± (12 + O (Λ))  β p ˆ γ − 1 Φ n 3 / 2 + κp ˆ γ − 1 Φ n 3 / 2  + O  p 2ˆ γ − 3 Φ 2 n 2  , where the O  1 / ( n 2 p )  error term w as absorb ed int o the O  n − 2 p 2ˆ γ − 3 Φ 2  -term since ˆ γ ≤ 1 and so 1 / ( n 2 p ) = o ( p 2ˆ γ − 3 Φ 2 /n 2 ). F urthermore, w e claim that one m ay n o w omit this latter error-term altogether as it is negligible compared to the error-term of O (Λ) in the terms in volving β , κ . I n deed, ke eping in mind that Φ and Λ − 1 are eac h of order log 2 n , w e ha ve p 2ˆ γ − 3 Φ 2 n 2 = O (Λ) p ˆ γ − 1 Φ n 3 / 2 log 4 n n 1 / 2 p 2 − ˆ γ ≤ O (Λ) p ˆ γ − 1 Φ n 3 / 2 log 4 n p n ( p ⋆ ) 4 − 2ˆ γ = O (Λ) p ˆ γ − 1 Φ n 3 / 2 log 4 n log 5 n = o  Λ p ˆ γ − 1 Φ n 3 / 2  . (3.18) Assume no w that X ( i ) ent ers the upp er critical interv al d efined b y I Y =  ˆ αn 1 / 2 p ˆ γ Φ , αn 1 / 2 p ˆ γ Φ  where ˆ α =  1 − log − 1 n  α . That is, supp ose that i 0 is the first round at whic h X ( i ) ≥ ˆ αn 1 / 2 p ˆ γ Φ and define the stopping time τ Y = min { i ≥ i 0 : X ( i ) < ˆ αn 1 / 2 p ˆ γ Φ } . As b efore, consid er Z ( i ) = X ( i ) − αn 1 / 2 p ˆ γ Φ . By the same argu m en t of (3.15) w e ha ve ∆ h αn 1 / 2 p ˆ γ Φ i =  6 + O ( n − 2 )   p ˆ γ − ˆ γ p ˆ γ − 1  αn − 3 / 2 Φ and com bined this with the ab ov e u pp er b ound on E [∆ X | F i ] establishes that E [∆ Z | F i , τ Y > i ] ≤  − 2 ˆ α + 2 β + 2 κ + ˆ γ α + O (Λ)  6 p ˆ γ − 1 Φ n 3 / 2 +  − α + O (Λ)  6 p ˆ γ Φ n 3 / 2 ≤  − (2 − γ ) α + 2 β + 2 κ  6 p ˆ γ − 1 Φ n 3 / 2 +  − α + O (Λ)  6 p ˆ γ Φ n 3 / 2 , where w e used the fact that γ − ˆ γ = 6 / log n to absorb b oth 2( α − ˆ α ) = 2 α/ log n and the O (Λ)-term for large n . In p articular, S j = Z (( i 0 + j ) ∧ τ Y ) is indeed a sup erm artin gale so long as Λ = o ( α ) and (2 − γ ) α ≥ 2 β + 2 κ . (3.19) It remains to b ound ∆ Z in L ∞ and L 2 . Here ther e are 2 typ es of roun ds: ones in which we choose a triangle that in v olv es u or v and a vertex in Y u,v (there are O ( n 2 p 4 ) such triangles) and ones where we c ho ose an y other triangle, in wh ich case Y u,v is unchange d . The f ormer even t h as probabilit y O ( p/n ) and leads to an O (1) c hange in Z while th e latter giv es a v ariation in Z of order O ( p/n ) due to the − 6 /n 2 c hange in p . Therefore, | ∆ Z | = O (1) and E [(∆ Z ) 2 | F i ] = O ( p/n ). Let p 0 = p ( i 0 ) = 1 − 6 i 0 /n 2 . By th e d efinition of i 0 and the f act that | ∆ Z | = O (1), Z ( i 0 ) ≤ ( ˆ α − α ) n 1 / 2 ( p 0 ) ˆ γ Φ + O (1) ≤ − 1 2 αn 1 / 2 ( p 0 ) ˆ γ log n (the last inequalit y holds for large n as the fi n al expression clearly tends to ∞ w ith n ), and therefore the sup ermartingale S j = Z  ( i 0 + j ) ∧ τ P  satisfies S 0 ≤ − 1 2 αn 1 / 2 ( p 0 ) ˆ γ log n , max j | S j +1 − S j | = O (1) , X j E  ( S j +1 − S j ) 2 | F ′ j  = O  n ( p 0 ) 2  , 13 where F ′ j = F i 0 + j . Sin ce n 1 / 2 ( p 0 ) 2 − ˆ γ ≥ c log 5 n for p 0 ≥ p ⋆ due to Eq. (2.11) we hav e | S 0 | max j | S j +1 − S j | = O  n ( p 0 ) 2  and so T heorem 3.1 yields th at for some fi xed c > 0 P ( ∪ j { S j ≥ 0 } ) ≤ exp  − c n ( p 0 ) 2ˆ γ log 2 n n ( p 0 ) 2  = exp  − c ( p 0 ) 2ˆ γ − 2 log 2 n  ≤ e − c log 2 n , whic h is suffi ciently small to sho w that w .h.p. X ( i ) < αn 1 / 2 p ˆ γ Φ for all u, v and i . The same argumen t handles the analog ous symmetric case of the critical in terv al  − αn 1 / 2 p ˆ γ Φ , − ˆ αn 1 / 2 p ˆ γ Φ  and sh o ws that w.h.p. X ( i ) > − αn 1 / 2 p ˆ γ Φ for all u, v and i . T his concludes the pro of of Eq. (2.6 ). 3.3 Edges within a neigh b orho o d ( T u ) The n umb er of edges in the subgraph ind uced b y the neigh b orho o d of u can c hange in tw o wa ys : Either a v ertex is remo ved from N u (due to selecting a triangle of the form uxy w ith x ∈ N u ) thereb y decremen ting T u b y all edges in ciden t to it in this ind uced subgraph, or N u remains u nc han ged (up on selecting a triangle that do es not include u ) and y et s ome of its in n er edges are remo v ed. The former case w ill b e handled b y directly su mming ov er x ∈ N u , noting there are Y ux triangles of th e form uxy wh er eas the v ertex x is inciden t to Y ux edges coun ted in T u . T he latter case requires a more delicate treatmen t, similar to the one used to study R u,v in Section 3.1. Indeed, the naive approac h wo u ld b e to sum o ver edges counte d b y T u , i.e. xy ∈ E w ith x, y ∈ N u , as eac h of these w ould decrease T u b y 1 up on selecting one of the Y x,y triangles in ciden t to it. Ho wev er, the cum u lative error in this app roac h (summing the co-degree errors for eac h edge in T u ) w ould b e quite substant ial as it completely ignores the effect of av er aging the co-degrees o v er T u . T o tak e adv an tage of this p oin t we will use our estimates for the random v ariables R x,u , whic h incorp orate this a verag ing effect. Namely , E [∆ T u | F i ] = − 1 Q X x ∈ N u Y 2 u,x + T u Q − 1 2 Q X x ∈ N u R x,u . (3.20) Here the first t wo terms accoun ted for triangles lost due to edges of the form ux (eac h is c h osen with probabilit y Y u,x /Q and eliminates Y u,x triangles f r om T u , hence the fir st term, y et in this wa y eac h xy ∈ T u is d ouble coun ted, hence the second correcting term). Th e last term counted triangles of the form xy z where x, y , z ∈ N u as w ell as ones of the form xy z wh ere x, y ∈ N u and z 6∈ N u ∪ { v } . F or a giv en x this corresp onds to R x,u (whic h we r ecall coun ts ordered suc h pairs ( y , z ), as needed since h a ving z ∈ N x,u w ould imp act t w o edges in T u ), and the final factor of 1 2 mak es up for the double coun t o ver all x ∈ N u . W e ev aluate the last term in (3.20) using the b oun ds (2.7) to get − 1 2 Q X x ∈ N u R x,u = − 1 2 Q X x ∈ N u  pY x Y x,u ± β n 3 / 2 p 2+ˆ γ Φ  = − p 2 Q X x ∈ N u ( Y x Y x,u ) ± (1 + O (Λ)) 3 β p ˆ γ Φ n 1 / 2 . (3.21) The fi rst sum in (3.20) can b e estimated b y Lemma 3.2 (noting that P x ∈ N u Y ux = 2 T u ), and so E [∆ T u | F i ] = − 1 Q  4 T 2 u Y u ± (4 + O (Λ)) α 2 n 2 p 1+2ˆ γ Φ 2  − p 2 Q X x ∈ N u ( Y x Y x,u ) ± (1 + O (Λ)) 3 β p ˆ γ Φ n 1 / 2 . (3.22) As usual set X ( i ) = T u − pY 2 u / 2 14 and consid er ∆  pY 2 u / 2  . Observ e that Y u c hanges if and only if the triangle selected is of the form uxy with x, y ∈ N u , in wh ic h case it decreases b y 2. Hence, E [∆ Y u | F i ] = − 2 T u /Q and ∆( Y 2 u ) = (2 Y u − 2)∆ Y u , and putting these toget h er we get E  ∆  pY 2 u / 2  | F i  = − 6 n 2 Y 2 u 2 − 1 2  p − 6 n 2  (2 Y u − 2) 2 T u Q = − 3 Y 2 u n 2 − 2 pT u Y u Q + O  p n  . Com binin g this estimate with (3.22 ) and th e b ound (2.8) for Y u giv es E [∆ X | F i ] = − 4 T u QY u  T u − p Y 2 u 2  − p 2 Q  X x ∈ N u Y x,u  np ± κn 1 / 2 p ˆ γ − 1 Φ   + 3 Y 2 u n 2 ± (1 + O (Λ))  3 β p ˆ γ Φ n 1 / 2 + 24 α 2 p 2ˆ γ − 2 Φ 2 n  By Eq. (3.18) we ha ve n − 1 p 2ˆ γ − 2 Φ 2 = o (Λ n − 1 / 2 p ˆ γ Φ) for all p ≥ p ⋆ , th us the ab o ve expression inv olving α 2 can b e absorb ed int o the O (Λ) error-term of the expression in vo lving β . F urthermore, sin ce 4 T u / ( QY u ) = (12 + O (Λ)) / ( n 2 p ) and p 2 Q  X x ∈ N u Y x,u  np ± κn 1 / 2 p ˆ γ − 1 Φ   − 3 Y 2 u n 2 = 6 n 2 p  T u − pY 2 u / 2  ± (3 + O (Λ)) κp ˆ γ Φ n 1 / 2 + O  p 2ˆ γ − 2 Φ 2 n  w e can conclude that E [∆ X | F i ] = − 18 + O (Λ) n 2 p X ± (1 + O (Λ)) 3( β + κ ) p ˆ γ Φ n 1 / 2 . No w we consider the upp er critical in terv al for T u giv en by I T =  ˆ µn 3 / 2 p 1+ˆ γ Φ , µn 3 / 2 p 1+ˆ γ Φ  where ˆ µ =  1 − log − 1 n  µ , and as b efore let i 0 b e the fir s t roun d in whic h X ( i ) ≥ ˆ µn 3 / 2 p 1+ˆ γ Φ, define the stopping time τ T = min { i ≥ i 0 : X ( i ) < ˆ µn 3 / 2 p 1+ˆ γ Φ } and consider Z ( i ) = X ( i ) − µn 3 / 2 p 1+ˆ γ Φ . Exactly the same argumen t of (3.15) gives ∆ h µn 3 / 2 p 1+ˆ γ Φ i =  6 + O ( n − 2 )   p 1+ˆ γ − (1 + ˆ γ ) p ˆ γ  µn − 1 / 2 Φ and together with the ab o ve up p er b ound on E [∆ X | F i ] we get E [∆ Z | F i , τ T > i ] ≤  − 6 ˆ µ + β + κ + (2 + 2 ˆ γ ) µ + O (Λ)  3 p ˆ γ Φ n 1 / 2 +  − µ + O (Λ)  6 p 1+ˆ γ Φ n 1 / 2 ≤  − (4 − 2 γ ) µ + β + κ  3 p ˆ γ Φ n 1 / 2 +  − µ + O (Λ)  6 p 1+ˆ γ Φ n 1 / 2 , where we used the fact that γ − ˆ γ = 6 / log n to absorb b oth 4( µ − ˆ µ ) = 4 µ/ log n and the O (Λ)-term for large n . In p articular, S j = Z (( i 0 + j ) ∧ τ Y ) is indeed a sup erm artin gale pro vided that Λ = o ( µ ) and (4 − 2 γ ) µ ≥ β + κ . (3.23) Ha ving established an L 1 b ound on ∆ S it remains to consider the corresp onding L 2 , L ∞ b ound s. If w e c ho ose a triangle of the f orm uxy , an eve nt that has probabilit y O (1 /n ), then T u decreases by Y ux + Y uy − 1 15 while Y u decreases b y 2, hence the c hange in Z in this case is at most the O ( n 1 / 2 p ˆ γ log 2 n ) due to th e error-terms in our appro ximation for the degrees and co-deg r ees. Th e probab ility that w e c h o ose a triangle that do es not con tain u y et includes an edge in T u is O ( p 2 ) and selecting such a triangle c hanges Z by O (1). The c hoice of an y other tr iangle c hanges Z by O ( p 2 ) due to the change in p . Alt ogether, | ∆ Z | = O  n 1 / 2 p ˆ γ log 2 n  and E [(∆ Z ) 2 | F i ] = O ( p 2ˆ γ log 4 n ). L et p 0 = p ( i 0 ) = 1 − 6 i 0 /n 2 and r ecall that the definition of i 0 and our b ound on | ∆ Z | ensure that Z ( i 0 ) ≤ ( ˆ µ − µ ) n 3 / 2 ( p 0 ) 1+ˆ γ Φ + O  n 1 / 2 ( p 0 ) ˆ γ log 2 n  ≤ − 1 2 µn 3 / 2 ( p 0 ) 1+ˆ γ log n , where the factor of 1 2 absorbs the O  n 1 / 2 ( p 0 ) ˆ γ log 2 n  -term since np 0 log − 1 n ≥ n 1 − o (1) p ⋆ > √ n for large enough n . I t th en follo w s the sup ermartingale S j = Z (( i 0 + j ) ∧ τ T ) satisfies S 0 ≤ − 1 2 µn 3 / 2 ( p 0 ) 1+ˆ γ log n , max j | S j +1 − S j | = O  n 1 / 2 p ˆ γ log 2 n  , X j E  ( S j +1 − S j ) 2 | F ′ j  = O  n 2 ( p 0 ) 1+2ˆ γ log 4 n  , where F ′ j = F i 0 + j . Here | S 0 | max j | S j +1 − S j | = o  n 2 ( p 0 ) 1+2ˆ γ log 4 n  due to one extra log f actor b etw een these expressions and therefore Theorem 3.1 establishes that f or some fixed c > 0 P ( ∪ j { S j ≥ 0 } ) ≤ exp  − c n 3 ( p 0 ) 2+2ˆ γ log 2 n n 2 ( p 0 ) 1+2ˆ γ log 4 n  = exp  − cnp 0 log − 2 n  ≤ e − √ n . W e conclude that w.h .p . X ( i ) < µn 3 / 2 p 1+ˆ γ Φ for all u and i , and the same argumen t s ho ws that w.h.p. X ( i ) > − µn 3 / 2 p ˆ γ Φ for all u and i . This concludes th e pro of of Eq. (2.9 ). 3.4 V ertex degrees ( Y u ) The analysis of the degrees will b e straigh tforward u sing our estimate (2.9) for T u , the num b er of inn er edges in the neigh b orho o d of a vertex u , sin ce Y u c hanges iff the triangle selected is of the form uxy (in whic h case it decreases by 2). Indeed, s etting X ( i ) = Y u − np , our b ound s on T u and Q imply that E [∆ X | F i ] = − 2 T u Q + 6 n = − pY 2 u ± 2 µn 3 / 2 p 1+ˆ γ Φ 1 6 n 3 p 3 ± α 2 n 2 p 2ˆ γ − 1 Φ 2 + 6 n = − p  X 2 + ( np ) 2 + 2 npX  1 6 n 3 p 3 + 6 n ± (12 + O (Λ)) µp ˆ γ − 2 Φ n 3 / 2 + O  p 2ˆ γ − 4 Φ 2 n 2  = − 12 n 2 p X ± (12 + O (Λ)) µp ˆ γ − 2 Φ n 3 / 2 , where the term pX 2 / ( n 3 p 3 ) was absorb ed in to the O ( p 2ˆ γ − 4 Φ 2 /n 2 )-term since | X | ≤ κ √ np ˆ γ − 1 Φ by (2.8), and this latter error-term w as thereafter omitted as it is o (Λ p ˆ γ − 2 Φ /n 3 / 2 ) b y (3.18). No w consider the upp er critical inte rv al for Y u giv en by I ′ Y =  ˆ κn 1 / 2 p ˆ γ − 1 Φ , κn 1 / 2 p ˆ γ − 1 Φ  where ˆ κ =  1 − log − 1 n  κ , 16 and as b efore assu me that i 0 is the first time at which X ( i 0 ) ≥ ˆ κn 1 / 2 p ˆ γ − 1 Φ and defi ne the stopping time τ ′ Y = min { i > i 0 : X ( i ) < ˆ κn 1 / 2 p ˆ γ − 1 Φ } . With th ese d efinitions, the v ariation in the v ariable Z ( i ) = X ( i ) − κn 1 / 2 p ˆ γ − 1 Φ consists of ∆ X as we ll as ∆  − κn 1 / 2 p ˆ γ − 1 Φ  ≤  6 + O ( n − 2 )  ( ˆ γ − 1) κp ˆ γ − 2 n − 3 / 2 Φ, hence E  ∆ Z | F i , τ ′ Y > i  ≤  − 2 ˆ κ + 2 µ + ( ˆ γ − 1) κ + O (Λ)  6 p ˆ γ − 2 Φ n 3 / 2 ≤  − (3 − γ ) κ + 2 µ  6 p ˆ γ − 2 Φ n 3 / 2 , where w e used the fact that γ − ˆ γ = 6 / log n eliminates the term 2( κ − ˆ κ ) = 2 κ/ log n and O (Λ)-term for large n . Hence, S j = Z (( i 0 + j ) ∧ τ ′ Y ) is a sup erm artin gale as long as (3 − γ ) κ ≥ 2 µ . (3.24) F urthermore, in eac h round w e either sel ect a triangle inciden t to u , an ev ent whic h has probabilit y O (1 /n ) and changes Z by O (1), or we do not affect Y u and thus c hange Z b y O (1 /n ) due to th e c hange in p . T h u s, | ∆ Z | = O (1) while E [(∆ Z ) 2 | F i ] = O (1 /n ), and w e conclude that for large enough n the sup ermartingale S j has the follo wing attributes: S 0 ≤ − 1 2 κn 1 / 2 ( p 0 ) ˆ γ − 1 log n , max j | S j +1 − S j | = O (1) , X j E  ( S j +1 − S j ) 2 | F ′ j  = O ( np 0 ) , where p 0 = p ( i 0 ) = 1 − 6 i 0 /n 2 and F ′ j = F i 0 + j . Since √ n ( p 0 ) 2 − ˆ γ ≥ √ n ( p ⋆ ) 2 − ˆ γ ≥ c log 5 n we dedu ce that | S 0 | max j | S j +1 − S j | = o ( np 0 ) and thus T heorem 3.1 establishes that for some fixed c > 0 P ( ∪ j { S j ≥ 0 } ) ≤ exp  − c n ( p 0 ) 2ˆ γ − 2 log 2 n np 0  = exp  − cp 2ˆ γ − 3 0 log 2 n  ≤ e − c log 2 n . Altoget h er, w.h.p . X ( i ) < κn 1 / 2 p ˆ γ − 1 Φ for all u and i , and similarly X ( i ) > − κn 1 / 2 p ˆ γ − 1 Φ w.h.p. for all u and i . Th is completes th e p ro of of Eq. (2.8). 3.5 Co-degree of triples ( Y u,v ,w ) W e will pr ov e th e f ollo win g result fr om whic h (2.10) will readily follo w: Y u,v ,w = np 3 ± q np 3 log 5 n for all u, v , w and p ≥ p 1 := n − 1 / 3 log 5 / 3 n . (3.25) Define X ( i ) = Y u,v ,w ( i ) − np 3 . W e hav e E [∆ X | F i ] = − X x ∈ N u,v ,w Y x,u + Y x,v + Y x,w − 1 { uv ∈ E } − 1 { uw ∈ E } − 1 { vw ∈ E } Q + 18 p 2 n ≤ − ( X + np 3 )(3 np 2 − αn 1 / 2 p ˆ γ Φ) n 3 p 3 / 6 + O  α 2 p 2ˆ γ − 2 Φ 2 n 2  + 18 p 2 n = − 18 n 2 p X + 6 αp ˆ γ Φ n 3 / 2 + O αp ˆ γ − 3 / 2 Φ log 2 n n 2 ! 17 No w su pp ose that i 0 is a firs t r ound at which X ( i 0 ) > 2 3 p np 3 log 5 n , that is X ente rs the critical inte r v al I ′′ Y =  2 3 q np 3 log 5 n , q np 3 log 5 n  . Setting τ ′′ Y = min { i > i 0 : X ( i ) < 2 3 p np 3 log 5 n } and Z ( i ) = X ( i ) − q np 3 log 5 n w e get E  ∆ Z | F i , τ ′′ Y > i  ≤ − 12 p 1 / 2 log 5 / 2 n n 3 / 2 + 6 αp ˆ γ Φ n 3 / 2 + O  αp ˆ γ − 3 / 2 Φ log 2 n n 2  + 9 p 1 / 2 log 5 / 2 n n 3 / 2 = 6 αp ˆ γ Φ − 3 p 1 / 2 log 5 / 2 n n 3 / 2 + O  αp ˆ γ − 3 / 2 Φ log 2 n n 2  . Since ˆ γ = 1 2 − O (1 / log n ) it follo ws that p ˆ γ Φ = O ( p 1 / 2 log 2 n ) = o ( p 1 / 2 log 5 / 2 n ) and hence the first term in th e ab o ve r.h.s. is equ al to − (3 + o (1))  p log 5 n n 3  1 / 2 . As for the second term th ere, recall that p ≥ p ⋆ and so b y (2.11) w e ha ve p 4 − 2ˆ γ ≥ 6 α 2 Φ 2 n − 1 log 6 n . In particular, p ˆ γ − 3 / 2 Φ log 2 n n 2 =  p log 5 n n 3  1 / 2 Φ n 1 / 2 p 2 − ˆ γ log 1 / 2 n ≤  p log 5 n n 3  1 / 2 1 √ 6 α log 7 / 2 n , and altoge ther we conclud e th at E  ∆ Z | F i , τ ′′ Y > i  ≤ − (3 + o (1)) p 1 / 2 log 5 / 2 n n 3 / 2 < 0 , where the last inequalit y holds an y for sufficien tly large n and confir ms that S j = Z (( i 0 + j ) ∧ τ ′′ Y ) is a sup ermartingale. Moreo ve r, th e equation that sp ecified E [∆ X | F i ] sho ws that if the tr iangle selected goes through u, v or w and a vertex in N u,v ,w , w hic h happ en s with pr ob ab ility O ( p 2 /n ), then the c hange in Z is O (1), and otherw ise the c hange in Z is O ( p 2 /n ). Hence, | ∆ Z | = O (1) while E [(∆ Z ) 2 | F i ] = O ( p 2 /n ), th us the sup ermartingale S j has the follo win g attributes: S 0 ≤ − 1 4 q n ( p 0 ) 3 log 5 n , max j | S j +1 − S j | = O (1) , X j E  ( S j +1 − S j ) 2 | F ′ j  = O  n ( p 0 ) 3  , where the factor 1 4 in the fir st expression (as opp osed to 1 3 ) treated the p oten tial O (1) deviation of Z ( i 0 ) from the lo wer endp oin t of the critical in terv al. Noting that p 0 ≥ p 1 =  log 5 n n  1 / 3 and hence | S 0 | max j | S j +1 − S j | = O  n ( p 0 ) 3  , an application of Theorem 3.1 yields that for some fixed c > 0, P ( ∪ j { S j ≥ 0 } ) ≤ exp  − c n ( p 0 ) 3 log 5 n n ( p 0 ) 3  = exp  − c log 5 n  . By the usual union b oun d o ver v ertices and roun ds we no w conclude that w.h.p. X ( i ) < p np 3 log 5 n for all u, v , w an d i , and similarly X ( i ) > − p np 3 log 5 n w .h.p. for all u, v , w and i , th us completing the pro of of Eq. (2.10). 18 3.6 Num b er of triangles ( Q ) In [5] it w as sho wn (see Th eorem 2 there) that Q ( i ) ≤ n 3 p 3 6 + 1 3 n 2 p throughout the pro cess w .h.p., hence it only remains to pro ve the lo wer b ound in (2.5). Let X ( i ) = Q − n 3 p 3 6 and recall that due to (2.2) w e ha ve E [∆ Q | F i ] = − X xy z ∈ Q Y x,y + Y x,z + Y y , z − 2 Q ≥ − 1 Q X xy ∈ E Y 2 x,y . T o b ound E [∆ X | F i ] from b elow w e will thus need an upp er b ound on P xy ∈ E Y 2 x,y . Recall that P xy ∈ E Y x,y = 3 Q and th at Y x,y = np 2 ± αn 1 / 2 p ˆ γ Φ by Eq. (2.6 ), h ence we can apply Lemma 3. 2 to- gether with the fact that | E ( i ) | = n 2 p/ 2 − n/ 2 to obtain that 1 Q X xy ∈ E Y 2 x,y ≤ 1 Q  9 Q 2 | E | + 4 | E | α 2 np 2ˆ γ Φ 2  ≤ 18 Q n 2 p  1 − 1 np  + 2 n 2 p  1 6 + O (Λ)  n 3 p 3 α 2 np 2ˆ γ Φ 2 ≤ 18 Q n 2 p + O ( p ) + (12 + O (Λ)) α 2 p 2ˆ γ − 2 Φ 2 = 18 Q n 2 p + (12 + O (Λ)) α 2 p 2ˆ γ − 2 Φ 2 , where in the last equalit y we absorb ed the O ( p )-term into th e O (Λ) err or-term factor of th e last expression since the facts ˆ γ ≤ 1 and ΛΦ 2 ≥ c log 3 n imply that p 2 − 2ˆ γ = o (ΛΦ 2 ), i.e. Λ p 2ˆ γ − 2 Φ 2 → ∞ . Add ing this to our estimate for E [∆ Q | F i ] while observing that ∆( − 1 6 n 3 p 3 ) = 3 np 2 + O ( p/n ) yields E [∆ X | F i ] ≥ − 18 Q n 2 p − (12 + O (Λ)) α 2 p 2ˆ γ − 2 Φ 2 + 3 np 2 + O ( p/n ) . As b efore we incorp orate the O ( p/n ) term in to the O (Λ) err or and using the defin ition of X w e can then rewrite the ab ov e as an upp er b ound on ∆( − X ), as f ollo ws: E [∆( − X ) | F i ] ≤ 18 n 2 p X + (12 + O (Λ)) α 2 p 2ˆ γ − 2 Φ 2 . No w assu me that i 0 is the first rou n d where X drops b elo w − ˆ α 2 n 2 p 2ˆ γ − 1 Φ 2 , i.e. en ters the in terv al I Q =  − α 2 n 2 p 2ˆ γ − 1 Φ 2 , − ˆ α 2 n 2 p 2ˆ γ − 1 Φ 2  where ˆ α =  1 − log − 1 n  1 / 2 α . F urther let τ Q = min { i > i 0 : X ( i ) > − ˆ α 2 n 2 p 2ˆ γ − 1 Φ 2 } and Z ( i ) = − X ( i ) − α 2 n 2 p 2ˆ γ − 1 Φ 2 . Since ∆  − α 2 n 2 p 2ˆ γ − 1 Φ 2  ≤  6 + O ( n − 2 )  (2 ˆ γ − 1) α 2 p 2ˆ γ − 2 Φ 2 , the upp er b oun d on ∆( − X ) gives E [∆ Z | F i , τ Q > i ] ≤  − 3 ˆ α 2 + 2 α 2 + (2 ˆ γ − 1) α 2 + O (Λ)  6 p 2ˆ γ − 2 Φ 2 ≤ − 12(1 − γ ) α 2 p 2ˆ γ − 2 Φ 2 , where the last inequalit y used the term γ − ˆ γ = 6 / log n to b oth cancel the O (Λ)-term and r eplace 3 ˆ α 2 b y 3 α 2 . As γ < 1 w e ded u ce th at S j = Z (( i 0 + j ) ∧ τ Q ) is indeed a sup ermartingale. Next consider the one-step v ariation of Z . Denoting the selected triangle in a given roun d by xy z , the c hange in Q follo win g this round is at m ost Y x,y + Y x,z + Y y , z and in ligh t of our co-degree estimate (2.6) 19 this expression d eviates from its exp ected v alue of 3 np 2 b y at most 3 αn 1 / 2 p ˆ γ log 2 n . In particular, | ∆ Z | = O  √ n ( p 0 ) ˆ γ log 2 n  and letting p 0 = p ( i 0 ) = 1 − 6 i 0 /n 2 this ensures that Z ( i 0 ) ≤ ( ˆ α 2 − α 2 ) n 2 ( p 0 ) 2ˆ γ − 1 Φ 2 + O  √ n ( p 0 ) ˆ γ log 2 n  ≤ − 1 2 α 2 n 2 ( p 0 ) 2ˆ γ − 1 log 3 n , where the last inequalit y holds for large n since n 3 / 2 ( p 0 ) ˆ γ − 1 log n tends to ∞ with n . 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