On $r$-colorability of random hypergraphs

On r -colorabilit y of random h yp ergraphs Andrey Kupa vskii ∗ , Dmitry A. Shabano v † Abstract The w ork deals with the threshold for r -colorabilit y in the binomial mo del H ( n, k , p ) of a random h yp ergraph. W e pro v e that if, for some constan t δ ∈ (0 , 1) , k ϕ ( k ) ln n ≪ r k − 1 ≤ n (1 − δ ) / 2 and p ≤ r k − 1 k − 1 − ϕ ( k ) n  n k  , where ϕ ( k ) is some func tion satisfying the relation ϕ ( k ) = Θ  q ln ln k ln k  , then P ( H ( n, k , p ) is r -colorable ) → 1 as n → ∞ . This result impro v es the previously kno wn results in the wide range of the parameters r = r ( n ) , k = k ( n ) . Keyw ords: r ando m hy p er gr aph , c olorings of hyp er gr aphs , sp arse hyp er gr aphs , r andom r e c oloring metho d . 1 In tro duction and history of the problem The w o r k deals with a proble m concerning threshold for r -colorabilit y in the binomial mo del of a random h yp ergraph. First of all, we recall the main definitions from the hypergraph theory . 1.1 Main definitions A hyp er gr aph H is a pair H = ( V , E ) , where V = V ( H ) is some finite set (c alled the vertex set of the hypergraph) and = E ( H ) is a collection of s ubsets of V , whic h are called the e dges of the h yp ergraph. If E ⊆  V k  , i.e. ev ery edge contains exactly k v ertices, then H is called k - uniform . By K ( k ) n w e denote a comple te k -uniform hypergraph on n v ertices. A v ertex coloring of the v ertex set V is calle d pr op er for h ypergraph H = ( V , E ) if in this coloring there is no mono c hromatic edges in E . A h ypergra ph H is called r -colorable, if there ∗ This work w as partially supported by the grant 09- 01-00 2 94 of Russian F oundation for Basic Resear c h and the gr ant MD -839 0.2010 .1 o f the Russian Pres iden t. † This work was partially supp orted by Russian F oundation of F undamental Resea rch (grant no. 09 - 01- 00294 ), by the pr o gram ” Leading Scientific Schools “ (gra nt no . NSh-878 4 .2010.1 ) and by the gra nt of Pre s iden t of Russian F ederation (no. MK-3 4 29.201 0.1). 1 exists a prop er coloring with r colors ( r - c oloring ) for H . The chr omatic numb er χ ( H ) of a h yp ergraph H is the minim um r such that H is r -colorable. Let v be a v ertex of a h yp ergraph H . T h e de gr e e of v in H is the n um b er of edges of H containin g v . By ∆( H ) w e denote the maxim um v ertex degree o f t he h ypergraph H . A h yp ergraph H = ( V , E ) is called l -simple , if e v ery t w o of its distinct e dges do not share more than l common v ertices , i.e. ∀ e, f ∈ E , f 6 = e : | e ∩ f | ≤ l . A 1-simple hypergraph is usually called simp le h ypergraph. A cycle of length 3 (3 - cycle or triangle ) in the h ypergraph H is a unordered set of three distinct edges ( e, f , h ) suc h that ( e ∩ f ) \ h 6 = ∅ , ( e ∩ h ) \ f 6 = ∅ , ( h ∩ f ) \ e 6 = ∅ . In this article w e study t he binomial mo del of a random hypergraph. Giv en in tegers n > k ≥ 2 and a real n um b er p ∈ [0 , 1] , r andom hyp er gr aph H ( n, k , p ) is a ra ndom spannin g subh ypergraph of the complete k -uniform h yp ergraph K ( k ) n with the follo wing distribution: for an y spanning subh yp ergraph F o f K ( k ) n , P ( H ( n, k, p ) = F ) = p | E ( F ) | (1 − p ) ( n k ) −| E ( F ) | . This definition immediatel y implies that ev ery edge of K ( k ) n is included in H ( n, k, p ) indep enden tly with equal probabilit y p . Supp ose Q n is a prop erty of k -uniform hy p ergraphs on n v ertices. W e sa y that Q n is an incr e asing pr op erty if, for an y t w o k -uniform h yp ergraphs H and G on n v ertices, H has prop erty Q n and E ( H ) ⊆ E ( G ) imply that G has prop erty Q n . F or giv en function k = k ( n ) ≥ 2 , the function p ∗ = p ∗ ( n ) is said to b e a thr eshold (or a thr eshold pr ob ability ) for an increasing prop ert y Q n , if • for a ny p = p ( n ) suc h that p ≪ p ∗ , P ( H ( n, k , p ) has prop erty Q n ) → 0 as n → ∞ ; • for a ny p = p ( n ) suc h that p ≫ p ∗ , P ( H ( n, k , p ) has prop erty Q n ) → 1 as n → ∞ . It follows f rom general results of B. Bollob´ as and A. Thomason (see [1]) conc erning monotone prop erties of random subsets that f o r a n y func tion k = k ( n ) ≥ 2 and any i ncreasing prop ert y Q n , the threshold probability exists. In this article w e are concen trated o n the estimating the threshold probabilit y fo r r -colora bilit y of H ( n, k , p ) , i.e. for an increasing prop ert y Q n = { h yp ergraph is not r -colorable } , where r = r ( n ) ≥ 2 is some function of n . In the next paragraph w e shall give a bac kground of thi s problem. 1.2 Previous resul ts The r -colorabilit y of random hypergraph H ( n, k , p ) was most inte nsiv ely studied in t he case of fixed k and r = 2 . It app ears that in this case the transition f rom 2-colorability to non-2- colorabilit y is sharp . It follows form the general results o f E. F reidgut (see [2]) that for any k ≥ 3 , the re exists a seq uence d k ( n ) suc h that fo r an y ε > 0 , • if p ≤ ( d k ( n ) − ε ) n/  n k  , then P ( H ( n, k , p ) is 2-colorable ) → 1 , 2 • but if p ≥ ( d k ( n ) + ε ) n/  n k  , then P ( H ( n, k , p ) is 2-color a ble ) → 0 . It is widely b eliev ed that d k ( n ) can b e re placed b y a constant d k . First b ounds for the threshold probabilit y of 2 - colorabilit y w ere obtained b y N. Alon and J. Sp encer. They show ed (see [3]) that there is a p ositiv e absolute constan t c suc h that if k ≥ k 0 is fixed and p ≤ c 2 k − 1 k 2 n  n k  , then P ( H ( n, k , p ) is 2 - colorable ) → 1 , (1) if k ≥ 3 , ε > 0 are fixed and p ≥ (1 + ε ) 2 k − 1 ln 2 n  n k  , then P ( H ( n, k , p ) is 2-colorable ) → 0 . (2) The g a p b et w een upp er and low er bounds in (1 ) and (2) w as reduced b y D. A c hlioptas, J.H. Kim, M. Kr ivele vic h and P . T etali from the order k 2 to the order k . They pro v ed (see [4]) that for an y fixed k ≥ 3 , if p ≤ 1 25 2 k − 1 k n  n k  , then P ( H ( n, k , p ) is 2 - colorable ) → 1 . (3) Finally , Ac hlioptas and C. Mo ore established (see [5]) the following b ound for the threshold probabilit y of 2-colorabilit y for all suffic ien tly large k : there is a constan t k 0 suc h that, for an y ε > 0 and any fixed k ≥ k 0 , if p ≤ (1 − ε ) 2 k − 1 ln 2 n  n k  , then P ( H ( n, k , p ) is 2 - colorable ) → 1 . (4) T ogethe r with the upper b ound (2) the inequalit y (4) giv es the exact v alue of the conside red sharp threshold for 2-colora bilit y in the case of fix ed k ≥ k 0 : p ∗ = 2 k − 1 ln 2 n  n k  . The r -colorability of H ( n, k , p ) f or r > 2 is not studied in suc h detail as 2-colora bility . The follo wing lemmas are just natura l generalizations of the results (1) and (2) of Alon and Sp encer. Lemma 1. Ther e exist p ositive c onstants C , c > 0 such that for any k = k ( n ) ≥ 3 and r = r ( n ) ≥ 2 , satisfying the c onditions r k − 1 /k ≥ C and r k − 1 = o ( n ) the fol low ing statement holds: if p ≤ c r k − 1 k 2 n  n k  , then P ( H ( n, k , p ) is r -c olor able ) → 1 . (5) Lemma 2. L et the functions k = k ( n ) and r = r ( n ) satisfy the r ela tion k 2 r = o ( n ) . Then for any fixe d ε > 0 , if p ≥ (1 + ε ) r k − 1 ln r n  n k  , then P ( H ( n, k , p ) is r -c olor able ) → 0 . (6 ) 3 Another result for r -colorability of random h ypergraphs w as obtained b y Ac hlioptas, Kim, Kriv elevic h and T etali. In the final commen t of their pap er [4] they stated (w ith pro vidi ng an algorithm of the pro of ) that the result (3) can b e generalized to the case of r colors in the follo wing form. Theorem 1. (D. Ac hlioptas, J.H. Kim, M. Krive levic h, P . T etali, [4]) Supp ose k ≥ 3 a n d r ≥ 2 ar e fixe d. If p = p ( n ) satisfies the ine quality p ≤ r ( r + 1)! ( r + 1) 2( r +1) r k − 1 k n  n k  , (7) then P ( H ( n, k , p ) is r-c olor able ) → 1 . It is easy to see that the b ound (5) of Lemma 1 is b etter than (7) if k ≤ c ( r + 1) 2( r +1) r ( r + 1)! = Ω ( r r ) . So, Theorem 1 g ives a new result o nly when r is small in comparison with k : r = O (ln k / ln ln k ) . The threshold probability for r -colorability of the r a ndom h yp ergr a ph H ( n, k , p ) in the case when r is large in comparison with k can b e obtained b y using the results concerning the c hromatic num b er of H ( n, k , p ) (recall, e.g., that (4), (7) a re nontrivial only when k is m uc h larger than r ). This problem w as studied by a series o f researc hers (see, e.g., [6], [7] for the bac kground). In o ur paper w e study H ( n, k , p ) in the “ sparse” case, i.e. the function p = p ( n ) is sufficie n tly small. F or suc h v alues of p , M. Kriv elevic h and B. Sudak ov pro v ed (see [7]) the follo wing theorem. Theorem 2. (M . Kriv elevic h, B. Sudak o v, [7]) L et k ≥ 3 b e fixe d. Ther e is a c o n stant d 0 = d 0 ( k ) such t hat, for any p = p ( n ) satisfying the c onditions d = d ( n ) = ( k − 1 )  n − 1 k − 1  p ≥ d 0 , d = o ( n k − 1 ) , the fol lowing c on v e r genc e holds : P  d k ln d  1 / ( k − 1) ≤ χ ( H ( n, k , p )) ≤  d k ln d  1 + 28 k ln ln d ln d  1 / ( k − 1) ! → 1 . One can make an immediate corollary from this the orem. Corollary 1. L et k ≥ 3 and ε ∈ (0 , 1) b e fi xe d. Ther e is a c onstant r 0 = r 0 ( k , ε ) such that, for any r = r ( n ) satisfying the c onditions r ≥ r 0 , r k − 1 ln r = o ( n k − 1 ) , the fol lowing c on v e r genc e holds : P ( χ ( H ( n, k , p )) ≤ r ) → 1 , wher e p = (1 − ε ) r k − 1 ln r n  n k  . 4 Corollary 1 together with Lemma 2 sho ws that the function p ∗ = r k − 1 ln r n/  n k  is a threshold pro ba bility in the wid e range: k is fixed, r is sufficien tly large in comparison w ith k and r k − 1 ln r = o ( n k − 1 ) . Ho w ev er, Theorem 2 (and, conseq uen tly , Corolla r y 1) can b e pro v ed not only for fixed k , but fo r slo wly gro wing functions k = k ( n ) also. The calcul ations from t he pro of of Theorem 2 pro vides the following neces sary re lations b et w een d , p , k and n : d ≥ (ln d ) 28 k − 27 , n 1 / 3 ≥  ln( n k − 1 p )  3( k − 1) − 1 / 2 . (8) These relations implies that in Corollary 1 w e hav e the fo llowing restrictions: r = Ω  k 29 (ln k ) 28  , n ≥ k 9 k + O ( k ln ln k / (ln k )) . (9) So, despite the fact that Corollary 1 giv es v ery go o d low er b ound for the threshold probabilit y , its statemen t holds only for large r in comparison with k : r = Ω ( k 29 (ln k ) 28 ) . Recall that (7 ) is b etter than (5 ) only when r = O (ln k / ln ln k ) . Henc e, in the very wide range o f the v alues of r , ln k ln ln k ≤ r ≤ k 29 (ln k ) 28 , (10) only the low er bound fro m Lemma 1 is known. R ema rk 1 . The pro of of Theorem 2 seems p ossible to b e adopted to the case o f s maller v alues of the par a meter d (and, consequen tly , parameter r in Corollary 1) than give n b y (8) . But, for example, the final condition r > k 4 seems to b e neces sary . So , the case when r is not very large in comparison with k is certainly not well studied. W e ha ve finished discussing previously know n res ults and now pro ceed to the new ones . 2 New results Our main approach of studying the threshold fo r r - color a bilit y of random h yp ergraph H ( n, k , p ) is to a pply metho ds and results concerning extremal problems of h yp ergraph coloring theory . 2.1 Colorings of hy p ergraphs with b ounded v ertex degrees F or all k , r ≥ 2 , let ∆( k , r ) denote the minim um p ossible ∆( H ) , where H is a k - uniform non- r -colorable hy p ergraph. The problem of finding or estimating the v alue ∆( k , r ) is one of the classical problems in extremal com binatorics. First b o unds for ∆( k , r ) w ere obtained b y P . Erd˝ os and L. Lov ´ asz (see [8]), they prov ed that for all k, r ≥ 2 , r k − 1 4 k ≤ ∆( k , r ) ≤ 20 k 2 r k +1 . (11) K osto c hk a and R¨ odl impro v ed (see [9]) the upp er b ound fro m (11). They show ed that for all k , r ≥ 2 , ∆( k , r ) ≤  k r k − 1 ln r  . 5 Classical lo w er bo und (1 1 ) of Erd˝ os and Lo v´ asz was improv ed b y J. Radhakrishnan and A. Sriniv asan (see [10]) in t he cas e r = 2 . They pro ved that for large n , ∆( k , 2) ≥ 0 . 17 2 k √ k ln k . Their result is still the best o ne in the c ase of t w o colors. When r > 2 D.A. Shabanov pro v ed (see [11]) a low er b ound with sligh tly b etter “p olynomial” factor: for any k ≥ 3 , r ≥ 3 , ∆( k , r ) > 1 8 k − 1 / 2 r k − 1 . (12) The last kno wn result concerning ∆( k , r ) w as recen tly obtained b y A.V. Kos to chk a, M. Kum bhat and V. R¨ odl (see [12]). They s how ed that if r = r ( n ) ≪ √ ln ln k , then ∆( k , r ) > e − 4 r 2  k ln k  ⌊ log 2 r ⌋ ⌊ log 2 r ⌋ +1 r k k . (13) The followin g lemma clarifies the connection b etw een the v a lue ∆( k , r ) and the threshold for r -colora bility of random h ypergraph H ( n, k , p ) . Lemma 3. S upp ose k = k ( n ) ≥ 2 and r = r ( n ) ≥ 2 satisfy the r elation 3 16 ∆( k , r ) − ln n → −∞ as n → ∞ . (14) If p ≤ 1 2 ∆( k , r ) k n  n k  , then P ( H ( n, k , p ) is r -c olor able ) → 1 . Pr o of. Since the probabilit y P ( H ( n, k , p ) is r -colorable ) decreases with growth of p , w e hav e to deal only with p = 1 2 ∆( k, r ) k n ( n k ) = 1 2 ∆( k , r )  n − 1 k − 1  − 1 . Let v b e a v ertex of H ( n, k , p ) and let X v denote the degree of v in H ( n, k , p ) . It is clear that X v is a binomial random v ariable B in  n − 1 k − 1  , p  . W e shall need a classical b ound on probability of large deviations fo r binomial v ariables (so called, Chernoff b ound): if X is a binomial random v ariable, then for an y λ > 0 , P ( X ≥ E X + λ ) ≤ exp  − λ 2 2( E X + λ/ 3)  . (15) The pro of of this classical fact can b e f ound, e.g., in the b o ok [13]. Using (15) with λ = E X w e get P  X v ≥ 2  n − 1 k − 1  p  = P ( X v ≥ ∆( k , r )) ≤ exp  − 3 16 ∆( k , r )  . Conseque n tly w e obtain the follo wing b ound for the probabilit y of the existence of the v ertices with large degree in H ( n, k , p ) : P (∆( H ( n, k , p )) ≥ ∆( k , r )) ≤ n ex p  − 3 16 ∆( k , r )  = exp  ln n − 3 16 ∆( k , r )  → 0 6 as n → + ∞ . The last relation follo ws from the condition (1 4). Th us, by the definition of the v alue ∆( k , r ) w e hav e lim n →∞ P ( χ ( H ( n, k , p )) ≤ r ) ≥ lim n →∞ P (∆( H ( n, k , p )) < ∆( k , r )) = 1 . Lemma 3 is prov ed.  As a corollary o f Lemma 3 and the b ounds (12) and (13 ) for ∆( k , r ) we immediately obtain the follow ing low er bound for the threshold for r -colorability of H ( n, k , p ) . Corollary 2. 1 ) S upp ose k = k ( n ) ≥ 3 and r = r ( n ) ≥ 3 satisfy the r elation 3 128 r k − 1 √ k − ln n → −∞ as n → ∞ . (16) If p ≤ 3 32 r k − 1 k 3 / 2 n  n k  , (17) then P ( H ( n, k , p ) is r -c olor able ) → 1 . 2) Supp ose k = k ( n ) ≥ 3 and r = r ( n ) ≥ 2 satisfy the r elation 3 16 e − 4 r 2  k ln k  ⌊ log 2 r ⌋ ⌊ log 2 r ⌋ +1 r k k − ln n → −∞ as n → ∞ . If r = o ( √ ln ln k ) and p ≤ 3 16 e − 4 r 2  k ln k  ⌊ log 2 r ⌋ ⌊ log 2 r ⌋ +1 r k k 2 n  n k  , (18) then P ( H ( n, k , p ) is r -c olor able ) → 1 . Let us compare the results of Corollary 2 with previous ones. Since Corollary 1 giv es almost complete answ er, w e hav e to compare (17) and (18) with (5) fro m Lemma 1 and (7) fr o m Theorem 1. Although the b ounds (5) and (7) formally hold only wh en k a nd r are fixed, the analogous statemen ts can b e prov ed b y using the same argumen ts for g r owing functions k = k ( n ) a nd r = r ( n ) . F or example, the analysis of the calculations in the papers [3], [4] and [5] sho ws that 1. The statemen t of Lemma 1 holds for almost a ll functions k = k ( n ) a nd r = r ( n ) , since it is true for r k − 1 = o ( n ) and in the case r k − 1 /k ≥ 22 ln n w e can just apply Lemm a 3 with the classical b ound (11) of Erd˝ os and Lo v´ asz. 2. The pro of of Theorem 1 can b e extended t o the fo llowing range of v alues of k = k ( n ) and r = r ( n ) : r is fixed and k = o ( √ n ) . 3. The pro of of the result (4) by Ac hlioptas and Mo ore do es not w ork for an y g rowing k = k ( n ) . It is also unclear ho w to generalize it to case o f fixed r > 2 . Th us, w e do not compare our new results with (4), since w e consider only the situation when k or r (or b oth of th em) is a gro wing function of n . 7 Let us sum up the obtained information. The lo w er b ound (5) from Lemma 1 holds almost for all r a nd k . Theorem 1 also can be extende d to a wide area of the v alues of the parameters. Ev erywh ere b elow for simplicit y w e sh all compare only the v alues of the bounds. Both (17) and (18) are ob viously b etter than (5). The second b ound (18) is w orse than (7). Indeed, the right hand-side of (7) is at least e − 2 r ln r r k − 1 k , whic h is b etter than (18 ), whose righ t hand-side is a t most e − 4 r 2 r k − 1 k ( ⌊ log 2 r ⌋ +2) / ( ⌊ log 2 r ⌋ +1) . The first b ound (1 7) of Corollary 2 is b etter than (7) i f √ k < 3 32 ( r + 1) 2( r +1) r ( r + 1)! . This inequalit y holds, e.g., w hen r ≥ ln k / ln ln k . Let us mak e interm ediate conclusions. Our new low er b ound (7) for the threshold probabilit y of r -colorability of random h yp ergraph H ( n, k , p ) impro v es all previously kno wn results in the follo wing wide area (see cond ition (9 ) of Corollary 1 and condition (16) of Corollary 2): ln k / ln ln k ≤ r ≤ k 29 (ln k ) 28 and r k − 1 √ k ≫ ln n. (19) W e see that in the area (19) the parameter r cannot b e v ery small in comparison with the n um b er of v ertices n , but it can b e v ery large. In the next paragraph w e shall presen t a b etter b ound when r is not very small and also is not very large in comparison with n . 2.2 Main result The main result o f our pap er is form ulated in the follo wing theorem. Theorem 3. Supp ose δ ∈ (0 , 1) i s a c onstant. L et k = k ( n ) and r = r ( n ) ≥ 2 satisfy the fol lowing c onditions: k ≥ k 0 ( δ ) , wher e k 0 ( δ ) is some c onstant, and, mo r e ove r, ( k − 1) ln r < 1 − δ 2 ln n, r k − 1 k − ϕ ( k ) ≥ 6 ln n, (20) wher e ϕ ( k ) = 4 j q ln k ln(2 ln k ) k − 1 . Then f o r function p = p ( n ) , satisfying p ≤ 1 2 r k − 1 k 1+ ϕ ( k ) n  n k  , (21) we have P ( H ( n, k , p ) is r -c olor able ) → 1 as n → ∞ . 8 Let us compare the re sult of Theorem 3 with previous ones. It is clear that the res triction (21) is w eak er (for all sufficien tly large k ) than our previous results (1 7) and (18) obtained in § 2.1. It is also obvious that (2 1) is b etter than the lo w er bo und (5) fro m Lemma 1 for all sufficien tly large k . So , it remains only to compare (21) with the result of Theorem 1 prov ed b y A c hlioptas, Kriv elevic h, Kim and T etali. W e see that (21) is asy mptotically b etter than (7) if k 4 j q ln k ln(2 ln k ) k − 1 < ( r + 1) 2( r +1) 2 r ( r + 1)! . This inequalit y holds, e.g., in the follo wing asymptotic area: r ≫ √ ln k . Th us, our main result (21) giv es a new lo w er b ound for the threshold probabilit y for r - colorabilit y of the random h yp ergraph H ( n, k , p ) . This new b ound improv es all the prev iously kno wn results in the wide area of the parameters (recall that we are workin g in the area (10)): √ ln k ≪ r ≤ k 29 (ln k ) 28 and 6 k ϕ ( k ) ln n ≤ r k − 1 ≤ n (1 − δ ) / 2 . F or example, (21) pro vides a new b o und when k ∼ r ∼ ln n/ (5 ln ln n ) . Moreov er, o ur result (21) is only k 1+ ϕ ( k ) ln r times smaller than the upp er bound (6) from Lemm a 2. The pro o f of Theorem 3 is based on some result concerning colorings of 2 -simple h yp ergraphs with b ounded v ertex degrees. The study of problems for colorings of simple h ypergraphs was initiated b y Erd˝ os a nd Lo v´ asz in [8]. Later the extremal problems concerning colorings of l - simple hypergraphs with b o unded v ertex degrees hav e b een con sidered b y Z. Sz ab´ o (se e [14]), A.V. Kostoch k a and M. Kum bhat (see [15]), D.A. Shabanov (see [16]). T o pro v e Theorem 3 w e consider 2-simple hy p ergraphs w ith a fe w 3- cycles . Let H b e an arbitrary h yp ergraph with the follo wing prop erties: H is k -uniform, χ ( H ) > r , H is 2- simple and for e v ery edge of H , there are at most ω 3-cycles con taining that edge . The class of all suc h hypergraphs we will denote by H ( k , r , ω ) . Let us consider the follow ing extremal v alue: ∆( H ( k , r, ω )) = min { ∆( H ) : H ∈ H ( k , r , ω ) } . Theorem 4 giv es an asymptotic lo wer b ound for ∆( H ( k , r , ω )) for ω = ⌊ p ln k / (ln ln k ) ⌋ . Theorem 4. Ther e exists an inte ger k 0 such that for al l k ≥ k 0 , al l r ≥ 2 an d al l ω ≤ p ln k / (ln ln k ) , ∆( H ( k , r, ω )) > r k − 1 k − 4 j q ln k ln(2 ln k ) k − 1 . (22) It should b e noted that the inequalit y (22) holds for all p o ssible v alues of the parameter r , whic h is imp orta n t fo r studying r -colorabilit y o f random h ypergra phs. F or a k -uniform, 2- simple, non- r -colorable h ypergra ph the lo w er bound for the maxim um v ertex degree similar t o (22) is kno wn o nly in the case of small r in comparison with k : r = O (ln k ) (see [16] for the details). The structure of the rest of the article will b e the followin g. In the next paragraph w e shall deduce Theorem 3 from Theore m 4. Section 3 will b e dev oted to the pro of of Theorem 4. Finally , in Se ction 4 w e s hall discuss c ho osabilit y in random hypergraphs. 9 2.3 Pro of of Theorem 3 Due to the decreasing o f t he probabilit y P ( H ( n, k , p ) is r -colorable ) with grow th of p w e ha v e to deal only with p = r k − 1 2 k 1+ ϕ ( k ) n ( n k ) . W e wan t to apply Theorem 4 to random h yp ergraph H ( n, k , p ) , so, w e ha v e to sho w that with probabilit y tending to 1 H ( n, k , p ) satisfies the follo wing conditions: it is 2-simple, eve ry edge is contained in at most ω = ⌊ p ln k / ln ln k ⌋ 3-cycles and, moreo v er, ∆( H ( n, k , p )) < ∆( H ( k , r, ω )) . Let v b e a v ertex of the ra ndom h yp ergraph H ( n, k , p ) and let X v denote the degree of v in H ( n, k , p ) . It is clear that X v is a binomial random v ariable B in  n − 1 k − 1  , p  . Using Chernoff b ound (15) with λ = E X v and the condition (20 ), we ha v e P  X v ≥ r k − 1 k − ϕ ( k )  ≤ exp  − 3 r k − 1 k − ϕ ( k ) / 16  ≤ exp {− (9 / 8) ln n } = n − 9 / 8 . Th us, P  ∆( H ( n, k , p )) ≥ r k − 1 k − ϕ ( k )  ≤ n · n − 9 / 8 = o (1) . (23) Let Y denote the num b er of pairs of edges, whose inte rsection has cardinality at least 3, and let Z denote the num b er of edges, whic h are con tained in a large n um b er, more than ω , of 3-cycles. W e es timate the expectations of these t w o random v ariables: E Y ≤  n 3  n − 3 k − 3  2 p 2 ≤ n 3 k 6 n 6  n k  2 r k − 1 k n  n k  − 1 ! 2 =  r k − 1  2 k 4 n . Conseque n tly , ln E Y = (2 k − 2) ln r + 4 ln k − ln n ( 20 ) ≤ (1 − δ ) ln n + 4 ln k − ln n ≤ − δ 2 ln n. Hence, lim n →∞ E Y = 0 and P ( H ( n, k , p ) is 2-simp le ) → 1 . (24) No w w e will consider edges, that are containe d in a large n um b er of triangles. Suppose u is an edge of H ( n, k , p ) . Let us denote b y T u the set of all triangles, con taining u . F urthermore, w e denote b y D ( u ′ , u ) the de gr e e of an e dge u ′ with r esp e ct to u , a n um b er o f 3-cycles from T u , con taining an edge u ′ 6 = u . Similarly , f o r an y v ertex v ∈ V ( H ( n, k , p )) , w e denote b y d ( v , u ) the the de gr e e of vertex v with r esp e ct to u , a n umber of triangles ( u , u ′ , u ′′ ) from T u suc h tha t v ∈ ( u ′ ∩ u ′′ ) \ u . No w w e will estimate the n um b er of edges that hav e big degree with respect to T u for some u . Denote b y Z 1 ( u ) the num b er of edges, ha ving degree greater than 4 with resp ect to u , i.e. Z 1 ( u ) = | { u ′ ∈ E ( H ( n, k , p )) : D ( u ′ , u ) > 4 }| . Moreo v er, let us de note Z 1 = P e ∈ E ( H ( n,k ,p ) Z 1 ( u ) . Now w e estimate the exp ectation of Z 1 : E Z 1 ≤ n  n − 1 k − 1  2 k 10  n − 2 k − 2  5 p 7 ≤  n k  7 k 22 n 11 r k − 1 k n  n k  − 1 ! 7 = r 7( k − 1) k 15 n 4 = 10 = exp { 7( k − 1) ln r + 15 ln k − 4 ln n } ( 20 ) ≤ exp  7 2 (1 − δ ) ln n + 15 ln k − 4 ln n  ≤ n − 1 / 2 . Let us explain the first inequalit y . A t first w e c ho ose the v ertex from the inters ection o f the edge u ′ with large degree and the edge u . Then w e c ho ose the rest v ertices of these tw o e dges. Then w e c hoo se 5 v ertices on b oth of edges, that corresp ond to remaining v ertices of fiv e 3-cycles. Then w e ch o ose th e last ed ge o f e ac h 3-cycle . Th us, P ( for an y u, u ′ ∈ E ( H ( n, k , p )) , D ( u ′ , u ) ≤ 4) → 1 . (25) Similarly , w e shall sho w, that with probabilit y tending to one, d ( v , u ) ≤ 4 for an y edge u and an y ve rtex v / ∈ u . Namely , w e denote b y Z 2 the n um b er of pairs v , u , such that d ( v , u ) ≥ 5 . Then the exp ectation o f Z 2 can b e estimated fro m ab o v e as fo llows. E Z 2 ≤ n  n − 1 k  k 5  n − 2 k − 2  5 p 6 ≤  n k  6 k 15 n 9 r k − 1 k n  n k  − 1 ! 6 = r 6( k − 1) k 9 n 3 = = exp { 6( k − 1) ln r + 9 ln k − 3 ln n } ( 20 ) ≤ exp { 3(1 − δ ) ln n + 9 ln k − 3 ln n } ≤ n − δ/ 2 . So, P ( for an y v ertex v a nd an edge u ∈ E ( H ( n, k , p )) , d ( v , u ) ≤ 4) → 1 . (26) Let us introduce the follo wing ev en t A n = { for any u , u ′ ∈ E ( H ( n, k , p )) and any v ∈ V ( H ( n, k , p )) , D ( u ′ , u ) ≤ 4 and d ( v , u ) ≤ 4 } . Due to (25) and (26 ) we ha v e that P ( A n ) → 1 . No w supp ose that the ev en t A n holds and there is an edge u in H ( n, k , p ) , whic h is containe d in at least ω 3-cycles. Consider the follo wing set of v ertices V u : V u = { v ∈ V : d ( v , u ) > 0 } . F or any v ∈ V u , b y E ( v , u ) w e denote the set o f edges, con taining v , whic h als o belongs to one of the 3- cycles from T u . The ev en t A n implies that d ( v , u ) ≤ 4 for any v and u a nd, hence, | E ( v , u ) | ≤ 8 and | V u | ≥ 1 8 | T u | = 1 8 ω . No w we will c onstruct a some con v enie n t subset of T u of sufficien t size. First, w e hav e V 0 u = V u and T 0 u = T u . Supp ose sets V s u and T s u are considered. W e fo rm a set V s +1 u and a set T s +1 u b y the follow ing w a y . W e c ho ose an arbitrary 3-cycle t s +1 = ( u ′ s , u ′′ s , u ) ∈ T s u and an arbitrary v s +1 ∈ ( u ′ s ∩ u ′′ s ) \ u . Then w e tak e V s +1 u = V s u \ { v s +1 } , T s +1 u = T s u \ { t ∈ T s u : t contains an edge from E ( v s +1 , u ) } . Then we rep eat the same pro cedure with sets T s +1 u , V s +1 u . The pro cedure con tin ues until both sets V s +1 u and T s +1 u are not empt y . Ho w man y steps of pro cedure can w e guaran tee? Since the ev en t A n holds, w e hav e that | T i +1 u | ≥ | T i u | − 3 2 . Indeed, E ( v i +1 , u ) consists of at most 8 edges and ev ery edge b elongs to at most 4 3-cycles. So, w e can g uar a n tee at least ω ′ = ⌈ 1 32 ω ⌉ steps of the pro cedure. 11 Consider the obtained set of 3-cycles { t 1 , . . . , t ω ′ } , t s = ( u ′ s , u ′′ s , u ) , s = 1 , . . . , ω ′ and the set of ve rtices { v 1 , . . . , v ω ′ } . O ur pro cedure shows that all the edges u ′ 1 , u ′′ 1 , u ′ 2 , u ′′ 2 , . . . , u ′ ω ′ , u ′′ ω ′ are distinct, all the v ertices v 1 , . . . , v ω ′ are also distinct and, for any s = 1 , . . . , ω ′ , we hav e v s ∈ u ′ s ∩ u ′′ s \ u . Let us estimate the probabilit y of the ev ent (denoted by B n ) that, for some edge u , the describ ed abov e c onfiguration of a pp ears in H ( n, k , p ) . It is cle ar th at P ( B n ) ≤  n k  k 2 ω ′ ( n − k ) ω ′  n − 2 k − 2  2 ω ′ p 2 ω ′ +1 ≤ n ω ′ k 6 ω ′ n 4 ω ′  n k  2 ω ′ +1 r k − 1 k n  n k  − 1 ! 2 ω ′ +1 = =  r k − 1  2 ω ′ +1 k 4 ω ′ − 1 n ω ′ − 1 . Hence, for all k ≥ k 0 , ln P ( B n ) = ( k − 1)(2 ω ′ + 1) ln r + (4 ω ′ − 1) ln k − ( ω ′ − 1) ln n ( 20 ) ≤ ( 20 ) ≤ ( ω ′ − 1)  (1 − δ )  2 ω ′ + 1 2 ω ′ − 2  ln n + 4 ω ′ − 1 ( ω ′ − 1) ln k − ln n  ≤ ( ω ′ − 1)  − δ 2 ln n  . (27) Conseque n tly , lim n →∞ P ( B n ) = 0 . Finally , if A n holds, then the ev en t that there is an edge u ∈ E ( H ( n, k , p )) with | T u | > ω implies the e vn t B n . Th us, P ( there is u ∈ E ( H ( n, k , p )) : | T u | > ω ) ≤ P  A n  + P ( B n ) → 0 . (28) Let us sum up the obtained information. It follo ws from (24) and (28) that with probability tending to 1 the random hypergraph H ( n, k , p ) satisfies all the conditions of being an elemen t of H ( k , r , ω ( k )) , ex cept the condition χ ( H ( n, k , p )) > r . Applying Theorem 4 and (23) we get that H ( n, k , p ) / ∈ H ( k , r, ω ( k )) with high probabilit y . Th us, P ( H ( n, k , p ) is r -colora ble ) → 1 as n → ∞ . Theorem 3 is pro ved. R ema rk 2 . If k = k ( n ) → + ∞ then the parameter δ ∈ (0 , 1) from Theorem 3 can b e tak en equal to some infinitesimal func tion. F or example, it follows from (27) that δ = 50 r ln ln k ln k is sufficien t. 3 Pro of of Theorem 4 The pro of of Theorem 4 is based on the method of random recoloring. This metho d in the case of tw o colors w as dev elop ed in the pap ers of J. Bec k [17], J. Sp encer [18 ], Radhakrishnan and Sriniv asan [10]. In this pap er w e follow the w ork [16] concerning r - color ings of l -simple h yp ergraphs with b ounded edge degrees. The structure of the pro of will b e the follo wing. In the next section w e will formulate a m ultiparametric Theorem 5 whic h pro vides a new low er b ound of the maximum edge degree in a h ypergra ph from the class H ( k , r, ω ) . Then w e will pro v e Theorem 5. Finally w e deduce Theorem 4 from Theorem 5 by c ho o sing the v a lues of the r equired paramete rs. 12 3.1 General theo rem Theorem 4 is a simple corollary of the follow ing multiparametric theorem. Theorem 5. L et k ≥ 3 , r ≥ 2 , ω ≥ 1 b e inte gers, let b , α b e p ositive numb ers. L et us denote: t = $ s ln k ln( α ln k ) % , q = α ln k k . (29) L et H = ( V , E ) b e an k -unifo rm 2-simple hyp er gr aph such that for eve ry e dge in H ther e ar e at most ω 3-cycles that c ontain that e dge. L et, mor e over, every e dge of h yp er gr aph H interse cts at most d other e dges of H , wher e d ≤ r k − 1 k 1 − b/t − 1 . (30) If the fol lowing ine qualities hold b ≤ t < k − ω , (31) 2 k ≤ q ≤ 1 2 , (32) k 2 2 k + ( t + 1 ) k 1 − α e α (ln k )( t + ω ) /k ( α ln k ) t + ω + ( t + 1) 2 t ! k 2 − b ( α ln k ) tω + + ( t + 1) t  2 eα ln k t − 1  t − 1 k 1+ α − b < 1 4 (33) then χ ( H ) ≤ r . The pro o f of this theorem is based on a metho d of v ertex ra ndom coloring. T o prov e Theorem 5 w e hav e to sho w the existenc e of a prop er v ertex r -coloring for h yp ergraph H . W e shall construct some random r - coloring and estimate the probabilit y that this coloring is not prop er for H . If this probability is greater than 0, the n w e prov e the existence of a req uired coloring, and the theorem follows . 3.2 Algorithm of random recol o ring W e follo w the ideas o f Radhak rishnan and Sriniv asan from [10] and the construction from [16] concerning random recoloring. Let V = { v 1 , . . . , v w } . The algor ithm consists of t w o phases. Phase 1. W e color all v ertices randomly and uniformly with r colors, independen tly from eac h other. Let us den ote the gene rated ra ndom coloring b y χ 0 . The obtained coloring χ 0 can contain mono c hromatic edges a nd “almost mono chromatic” edges. An edge e ∈ E is s aid to be alm ost m ono c h r omatic in χ 0 if there is a color u suc h that n − t − ω + 2 ≤ |{ v ∈ e : v is colored by u in χ 0 }| < n. In this case, the color u is called dom inating in e . F or ev ery v ∈ V , u = 1 , . . . , r , let us use the notations M ( v ) = { e ∈ E : v ∈ e, e is monochromatic in χ 0 } , AM ( v , u ) = { e ∈ E : v ∈ e, e is almos t mono c hromatic in χ 0 with dominating color u } . 13 Phase 2. In this phase, w e w an t to recolor some v ertices from the edges, which are mono c hromatic in χ 0 . W e consi der the vertic es according to an a rbitrary fixed order v 1 , . . . , v w . Let { η 1 , . . . , η w } be m utually indep enden t equally d istributed random v ariables, taking v alues 1 , . . . , r with the same proba bility p (the v alue of the parameter p will b e c hosen later) and the v alue 0 w ith probability 1 − r p . The re coloring pro cedure cons ists of w steps. Step 1. Assume that M ( v 1 ) 6 = ∅ and, moreo v er, there is no u = 1 , . . . , r a nd e ∈ AM ( v 1 , u ) suc h that (a) η 1 = u , (b) v 1 is the only verte x in e , whic h is not colored by u in χ 0 . Then w e try to recolor v 1 according to th e v alue of the random v ariable η 1 : • if η 1 = 0 , then we do not recolor v 1 , • if η 1 6 = 0 , then we recolor v 1 in the color η 1 . In all the other situations, we do not c hange the color of v 1 . Let χ 1 b e the coloring after considering v 1 . No w let the ve rtices v 1 , . . . , v i − 1 ha v e b een considered, so that the coloring χ i − 1 is obtained. Step i. Assume that some f ∈ M ( v i ) is still mono c hromatic in χ i − 1 and, moreo v er, there is no u = 1 , . . . , r and e ∈ A M ( v i , u ) suc h that (a) η i = u , (b) v i is the only verte x in e , whic h is not colored b y u in χ i − 1 . Then w e try to recolor v i according to the v alue of the random v ariable η i : • if η i = 0 , then we do not recolor v i , • if η i 6 = 0 , then we recolor v i in the color η i . In all the o t her situations, w e do not c hange the color of v i . Let the resulting coloring b e χ i . Let ˜ χ be the coloring obtained after the consideration o f all the vertic es. No w we are going to giv e a more formal construction of the random coloring ˜ χ using the tec hniques of random v ariables. This is v ery useful for the further proo f. W e analyze the ev ent F that ˜ χ is not a prop er coloring for H . W e divide F in to some “lo cal” ev en ts and estimate their probabilities. F inally , we use Lo cal Lemma to sho w that all these ev en ts do not o ccur sim ultaneously with p ositiv e probabilit y . This implies that ˜ χ is a prop er coloring of H with p ositiv e probabilit y , and, hence, H is r -colora ble. 14 3.3 F ormal Construction of the random coloring from § 3.2 Without loss of generalit y , w e ma y assume, that V = { 1 , 2 , 3 , . . . , w } . Let us also fix an arbitrary ordering o f the edges of H . Consider, on some probabilit y space, the follo wing set of m utually indep enden t random elemen ts: 1. ξ 1 , . . . , ξ w — equally distributed random v ariables, taking v alues 1 , 2 , . . . , r with equal probabilit y 1 /r . 2. η 1 , . . . , η w — equally distributed random v ariables taking v alues 1 , 2 , . . . , r with equal probabilit y p and the v alue 0 with probabilit y 1 − r p . W e t a k e the parameter p equal to p = q / ( r − 1) . By the condition (32) suc h choice of t he parameter is correct, i. e., for ev ery r ≥ 2 , one has th e inequalities rp ≤ r / (2( r − 1 )) ≤ 1 . Let e ∈ E b e an edge of H . F or e v ery u = 1 , . . . , r , let M ( e, u ) and AM ( e, u ) denote the follo wing ev en ts: M ( e, u ) = \ s ∈ e { ξ s = u } , AM ( e, u ) = ( 0 < X s ∈ e I { ξ s 6 = u } ≤ t + ω − 2 ) . (34) W e shall introduce success iv ely random v ariables ζ i , i = 1 , . . . , w . Let D 1 and A 1 denote the follo wing ev en ts: D 1 = [ e ∈ E : 1 ∈ e r [ u =1 M ( e, u ) , A 1 = [ f ∈ E : 1 ∈ f r [ u =1 ( ξ 1 6 = u, η 1 = u, X s ∈ f : s> 1 I { ξ s = u } = k − 1 ) ∩ AM ( f , u ) ! , and let ζ 1 = ξ 1 I { D 1 ∪ { η 1 = 0 } ∪ A 1 } + η 1 I {D 1 ∩ { η 1 6 = 0 } ∩ A 1 } . F or ev ery i > 1 , let D i and A i denote the ev en ts D i = [ e ∈ E : i ∈ e r [ u =1 ( M ( e, u ) ∩ \ s ∈ e : si I { ξ s = u } = k − 1 ) ∩ AM ( f , u ) ! . W e define ζ i in the follow ing wa y: ζ i = ξ i I { D i ∪ { η i = 0 } ∪ A i } + η i I {D i ∩ { η i 6 = 0 } ∩ A i } . It is easy to see that the random v ariables ζ i tak e v alues only fro m { 1 , 2 , . . . , r } . So, w e ma y in terpret the random v ector ~ ζ = ( ζ 1 , . . . , ζ w ) as a random r -coloring of t he v ertex set V (w e assign the color ζ i to the v ertex i ). Let F denote the ev en t that ~ ζ is not a prop er coloring of the h yp ergraph H , i. e., F = [ e ∈ E r [ u =1 \ s ∈ e { ζ s = u } . (35) 15 Our task is to prov e that P ( F ) < 1 under the conditions of Theorem 5. W e s hall div ide the ev en t T s ∈ e { ζ s = u } into three parts, dep ending on the b ehavior of the random v ariables { ξ s : s ∈ e } . Let C 0 ( e, u ) , C 1 ( e, u ) , C 2 ( e, u ) b e the follo wing ev en ts: C 0 ( e, u ) = r [ a =1 , a 6 = u \ s ∈ e { ζ s = u, ξ s = a } , C 1 ( e, u ) = \ s ∈ e { ζ s = u, ξ s = u } , C 2 ( e, u ) = \ s ∈ e { ζ s = u } ∩ r \ a =1 M ( e, a ) . (36) W e shall consider t hese ev en ts separately . But b efore we establish a simple inequalit y , whic h w e will use later. It f ollo ws from (32) that α ln k = q k ≥ 2 . (37) Note that the last inequalit y i mplies that the parameter t in (29) is correctly de fined (there is no negativ e num b er under the square ro o t). 3.4 First pa rt o f F : the e v en t C 0 ( e, u ) If the eve n t C 0 ( e, u ) o ccurs, the n fo r ev ery s ∈ e , one has ζ s = η s , since ζ s 6 = ξ s . W e get the relation r [ u =1 C 0 ( e, u ) ⊂ r [ u =1 r [ a =1 , a 6 = u \ s ∈ e { η s = u, ξ s = a } = Q 0 ( e ) . (38) The probabilit y of th e ev en t Q 0 ( e ) can b e e asily calculated: P ( Q 0 ( e )) = r X u =1 r X a =1 , a 6 = u Y s ∈ e P ( η s = u, ξ s = a ) = r ( r − 1)  p r  k . (39) 3.5 Second part o f F : the ev e nt C 1 ( e, u ) Supp ose that the ev en t C 1 ( e, u ) o ccurs. This ev en t, ob viously , implies all the ev en ts D s , s ∈ e . Then the equality ξ s = ζ s = u for a v ertex s ∈ e can happen in t w o w a ys: either η s ∈ { 0 , u } , or η s / ∈ { 0 , u } and the eve n t A s o ccurs. Consider the fo llo wing partition of the ev ent C 1 ( e, u ) : C 1 ( e, u ) = S 0 ( e, u ) ∪ S 1 ( e, u ) , (40) where S 0 ( e, u ) = C 1 ( e, u ) ∩ ( X s ∈ e I { η s / ∈ { 0 , u } } ≤ t + ω − 1 ) , S 1 ( e, u ) = C 1 ( e, u ) ∩ ( X s ∈ e I { η s / ∈ { 0 , u }} > t + ω − 1 ) . Consider the ev en t S 0 ( e, u ) . By the definition (36) of the ev en t C 1 ( e, u ) the following relation holds: S 0 ( e, u ) ⊂ \ s ∈ e { ξ s = u } ∩ ( X s ∈ e I { η s / ∈ { 0 , u }} ≤ t + ω − 1 ) . 16 Let Q 1 ( e ) denote the un ion o f the last ev en ts: r [ u =1 S 0 ( e, u ) ⊂ r [ u =1 ( \ s ∈ e { ξ s = u } ∩ ( X s ∈ e I { η s / ∈ { 0 , u }} ≤ t + ω − 1 )) = Q 1 ( e ) . (41) The probabilit y of Q 1 ( e ) has the follo wing estimate: P ( Q 1 ( e )) = r 1 − k t + ω − 1 X j = 0  k j  q j (1 − q ) k − j ≤ r 1 − k (1 − q ) k − t − ω t + ω − 1 X j = 0 ( k q ) j ≤ ≤ r 1 − k (1 − q ) k − t − ω ( k q ) t + ω . (42) The last inequalit y follows from the b ound (37): k q = α ln k ≥ 2 . Consider no w the ev en t S 1 ( e, u ) . Let us fix v ∈ e satisfying η v / ∈ { 0 , u } . As it w as noted ab o v e, the eve n t A v should happ en for ev ery suc h v ertex. This ev ent implies that for some edge f , v ∈ f , and some c olor a 6 = u , the follow ing ev en t has to o ccur W ( v , f , u, a ) = ( ξ v = u, η v = a, X s ∈ f : sv I { ξ s = a } = k − 1 ) ∩ AM ( f , a ) . It is easy to sho w that f 6 = e, moreo v er, f ∩ e = { v } . Indeed, for all s ∈ e , it holds that ξ s = ζ s = u , but for all s ∈ f \{ v } , either ζ s = a , or ξ s = a . Supp ose { v 1 , . . . , v h } = { v ∈ e : η v / ∈ { 0 , u }} . F or an y i = 1 , . . . , h , the ev en t S 1 ( e, u ) implies the ev en t W ( v i , f i , u, a i ) for some edge f i satisfying { v i } = f i ∩ e and some color a i 6 = u . Moreo v er, S 1 ( e, u ) also implies that h = h ( e, u ) ≥ t + ω . Since there a re at most ω 3-cycles con taining e , there is a subs et { f ′ 1 , . . . , f ′ t } ⊂ { f 1 , . . . , f h } suc h tha t f ′ i and f ′ j are disjoin t for all i 6 = j . F or furthe r conv enience, w e in tro duce a notatio n of the c onfigur a tion of the first typ e . F or giv en edge e , the set of ed ges { f 1 , . . . , f t } is said to b e the configuration of the first ty p e (denotation: { f 1 , . . . , f t } ∈ CONF1( e ) ) if, for an y any i = 1 , . . . , t , | f i ∩ e | = 1 and, moreov er, all the edges f i are pairwise disjoint. Th us, b y the ab ov e argumen ts w e t he follo wing relation S 1 ( e, u ) ⊂ \ s ∈ e { ξ s = u } ∩ r [ a 1 ,..., a t =1 a i 6 = u [ { f 1 ,..., f t }∈ CONF1( e ) t \ i =1 W ( e ∩ f i , f i , u, a i ) , (43) where the set of edges { f 1 , . . . , f t } is assumed to be or dered according to the originally selected ordering of E , i. e . the n um b er of the edge f i is less th an the n umber o f the edge f j , if i < j . Let us u se the notations: b f i = f i \ e and v i = e ∩ f i , i = 1 , . . . , t . It follo ws from the definition of the configuration of the first t ype that the sets b f i , i = 1 , . . . , t do not ha v e common vertice s, i.e. b f i ∩ b f j = ∅ , if i 6 = j . F urthermore, | b f i | = k − 1 . If the ev en t W ( e ∩ f i , f i , u, a i ) happens, then by AM ( f i , a i ) the edge f i con tains at most t + ω − 2 v ertices s , satisfying ξ s 6 = a i . Moreov er, for all suc h v ertices, ζ s = a i , and so, ζ s = η s = a i . 17 The set b f i con tains at most t + ω − 3 such v ertices, since the v ertex v i do esn’t belong to b f i and ξ v i = u 6 = a i . Th us, we obtain the relation \ s ∈ e { ξ s = u } ∩ t \ i =1 W ( e ∩ f i , f i , u, a i ) ⊂ \ s ∈ e { ξ s = u } ∩ t \ i =1 { η v i = a i }∩ ∩ t \ i =1    \ s ∈ b f i ( { ξ s 6 = a i , η s = a i } ∪ { ξ s = a i } )    ∩ t \ i =1    X s ∈ b f i I { ξ s 6 = a i } ≤ t + ω − 3    . (44) Let Q 2 ( e, F ) denote the union o f the last ev en ts, whe re F = { f 1 , . . . , f t } ∈ CONF1( e ) : Q 2 ( e, F ) = r [ u =1 r [ a 1 ,..., a t =1 a i 6 = u ( \ s ∈ e { ξ s = u } ∩ t \ i =1 { η v i = a i }∩ ∩ t \ i =1    \ s ∈ b f i ( { ξ s 6 = a i , η s = a i } ∪ { ξ s = a i } )    ∩ t \ i =1    X s ∈ b f i I { ξ s 6 = a i } ≤ t + ω − 3       . (45) The relations (43) and (44 ) imply r [ u =1 S 1 ( e, u ) ⊂ [ F ∈ CONF1( e ) Q 2 ( e, F ) . (46) Let us estimate the proba bility of Q 2 ( e, F ) : P ( Q 2 ( e, F )) = r X u =1 r X a 1 ,..., a t =1 a i 6 = u r − k p t t Y i =1 t + ω − 3 X j = 0  | b f i | j   r − 1 r  j p j  1 r  | b f i |− j = = r ( r − 1 ) t r − k p t r − t P i =1 | b f i | t Y i =1 t + ω − 3 X j = 0  | b f i | j  q j = r ( r − 1 ) t r − k p t r − t ( k − 1) t Y i =1 t + ω − 3 X j = 0  k − 1 j  q j ≤ ≤ r − ( t +1)( k − 1) q t t Y i =1 t + ω − 3 X j = 0 k j q j ≤ r − ( t +1)( k − 1) q t ( k q ) t ( t + ω − 2) . (47) 3.6 Third part of F : the e v en t C 2 ( e, u ) W e shall sho w that if the ev en t C 2 ( e, u ) happ ens then the sum P s ∈ e I { ξ s 6 = u } cannot b e v ery small. W e shall establish the equality C 2 ( e, u ) = C 2 ( e, u ) ∩ ( X s ∈ e I { ξ s 6 = u } ≥ t + ω − 1 ) . (48) 18 Indeed, let us consider the in tersection of three eve n ts (see the defi nition of t he ev en t C 2 ( e, u ) in (36)): C 2 ( e, u ) ∩ ( X s ∈ e I { ξ s 6 = u } ≤ t + ω − 2 ) = = \ s ∈ e { ζ s = u } ∩ r \ a =1 M ( e, a ) ∩ ( X s ∈ e I { ξ s 6 = u } ≤ t + ω − 2 ) . The second and the third ev en ts imply the happ ening of the ev en t AM ( e, u ) (see (34)). The first one implies that for ev ery s ∈ e satisfying ξ s 6 = u , w e hav e ζ s = η s = u . Moreo v er, since the ev en t AM ( e, u ) holds, the set of suc h v ertices in not empt y . Consider a v ertex v ∈ e satisfying ξ v 6 = u and ξ s = u for ev ery s ∈ e, s > v . It is clear that the e v en t A v holds. So, ζ v = ξ v 6 = u , and we get a con tradiction with the first eve n t in the inters ection. Th us, thes e three ev ents are inconsisten t, and w e prov e the equalit y (48). Let us estimate the probability of C 2 ( e, u ) . Consider the random set T = { s ∈ e : ξ s 6 = u } . The ev en t C 2 ( e, u ) implies, first, t ha t a ll v ∈ T s atisfy ζ v = η v = u , and second, that | T | ≥ t + ω − 1 (see (48)). Le t u s use the d enotation: E ( e ) = { f ∈ E \{ e } : f ∩ e 6 = ∅} . If ζ v 6 = ξ v for some v ertex v , then there should happ en at least t w o ev ents: the ev ent D v and the ev en t B ( e, f v , v , u, a v ) = ( M ( f v , a v ) ∩ \ s ∈ f v : s 0 . But in this case H j consists of only one edge f . Indeed, if g ∈ E ( H j ) , g 6 = f , then | g ∩ e | = 1 and, moreo v er, | g ∩ f | > 0 . This can only happ ens when f ∩ e = g ∩ e , i.e. g has the same first v ertex in its in tersection with e as f . This fact is in conflict with the definition of the configuration of the second type. So, G j is just an isolated v ertex. No w let h j > 0 . Sinc e H j is connected a nd F ∈ CONF2( e ) , G j is also connected. Suppo se there is a cycle ( w 1 , . . . , w m ) , m ≥ 3 , in G j , i.e. { w i , w i +1 } ∈ E ( G j ) , i = 1 , . . . , m − 1 and, moreo v er, { w 1 , w m } ∈ E ( G j ) . Without loss o f generalit y , assume that w 1 < w j for an y j > 1 , i.e. w 1 is the v ertex with the least n um b er in the cycle. Since { w 1 , w 2 } ∈ E ( G j ) , there is an edge g 1 ∈ E ( H j ) suc h that { w 1 , w 2 } = g 1 ∩ e , so w 1 is the first v ertex in g 1 ∩ e . By analo g y , there is another edge g 2 ∈ E ( H j ) s uc h that { w 1 , w m } = g 2 ∩ e , so w 1 is also the first v ertex in g 2 ∩ e . W e obtain a contradic tion with the fact that F ∈ CONF2( e ) . Henc e, G j is a tree.  Claim 1 implies that | V ( G j ) | = | E ( G j ) | + 1 = h j + 1 , th us, | S ( F ) | = l X j = 1 | V ( G j ) | = l X j = 1 ( h j + 1) . (56) Moreo v er, from the d efinitions of the v alues h j and l j w e get that, for an y j = 1 , . . . , l ,       [ f ∈ H j f       = ( k − 2) h j + ( k − 1) l j + | V ( G j ) | = ( k − 1)( h j + l j ) + 1 . (57) 21 Finally , Claim 1 implies that, for an y j = 1 , . . . , l , l j ≤ 1 . (58) Indeed, since there is a bijec tion b et w een the edges of H j and the first v ertices in their in tersection with e , we ha v e h j + l j = | E ( H j ) | ≤ | V ( G j ) | = h j + 1 , and the inequalit y (58 ) follow s. Using the notations introduced ab o v e one can easily find the probabilit y of the ev en t Q 3 ( e, F ) (see (53)): P ( Q 3 ( e, F )) = r  1 r + q r  k −| S ( F ) | p | S ( F ) | ( r − 1) l l Y j = 1 r −    S f ∈ H j f    . (59) Let us e xplain the last t w o factors in the righ t-hand side of (59). Sinc e all the edges of F are mono c hromatic in t he main coloring ξ , the v alues of ξ s should coincide for all s ∈ V ( H j ) . Th us, w e only hav e to c hoose a color (n ot equal to u ) for ev ery comp onen t (the fa ctor ( r − 1) l ). The last fa ctor is equal to the probabilit y (w e ha v e already chosen the colors) that ev ery edge in the comp onen t H j is mono c hromatic in the main coloring ξ . Using obtained relations (55), (56), (57), (5 8), we get the fo llowing estimate of the probabilit y of the ev en t Q 3 ( e, F ) : P ( Q 3 ( e, F )) = r  1 r + q r  k −| S ( F ) | p | S ( F ) | ( r − 1) l l Y j = 1 r −    S f ∈ H j f    = = r 1 − k (1 + q ) k −| S ( F ) | ( r p ) | S ( F ) | ( r − 1) l l Y j = 1 r − ( k − 1)( h j + l j ) − 1 = = r 1 − k (1 + q ) k −| S ( F ) | ( r p ) | S ( F ) | ( r − 1 ) l r − ( k − 1)( t − 1) − l ≤ ≤ r (1 − k ) t (1 + q ) k ( r p ) | S ( F ) | ≤ r (1 − k ) t (1 + q ) k (2 q ) t − 1 . (60) W e need to commen t o nly the last inequalit y . F rom the condition (32) w e ha v e 2 q ≤ 1 and, moreo v er, we kno w that r p = ( r / ( r − 1)) q ≤ 2 q . Finally , f r o m (55), (57) and (58) we immediately see that | S ( F ) | ≥ t − 1 . The b ound (6 0) completes the estimation of differen t parts of the ev en t F . No w w e shall pro v e that the probabilit y of F is less than 1 under the conditions of Theorem 5. 3.7 Application of Lo cal Lemma for estimating the probabilit y of F Remem ber that b y the definitions ( 3 5) and (36) of the ev en ts F and C i ( e, u ) , i = 1 , 2 , 3 , e ∈ E , u = 1 , . . . , r , w e hav e the equalit y F = [ e ∈ E r [ u =1 ( C 1 ( e, u ) ∪ C 2 ( e, u ) ∪ C 3 ( e, u )) . 22 It follo ws from t he obtained relations (38), (40), (41), (46) and (54), that F ⊂ [ e ∈ E {Q 0 ( e ) ∪ Q 1 ( e ) } ∪ [ e ∈ E [ F ∈ CONF1( e ) Q 2 ( e, F ) ∪ [ e ∈ E [ F ∈ CONF2( e ) Q 3 ( e, F ) . (61) F urther, w e shall use a classical claim, whic h is called Lo cal Lemma. This statemen t w as first pro v ed in the pap er of P . Erd˝ os and L. Lov´ asz [8]. W e shall form ulate it in a sp ecial case. Theorem 6. L et even ts B 1 , . . . , B N b e given on so m e pr ob ability sp ac e. L et S 1 , . . . , S N b e subsets of R N = { 1 , . . . , N } such that for any i = 1 , . . . , N , the event B i is indep end ent o f the algebr a gener ate d by the events {B j , j ∈ R N \ S i } . I f, for any i = 1 , . . . , N , the fol low ing ine quali ty holds X j ∈ S i P ( B j ) ≤ 1 / 4 , (62) then P N T j = 1 B j ! ≥ N Q j = 1 (1 − 2 P ( B i )) > 0 . The pro of of the Lo cal Lemma can b e found in the b o o k [19]. Consider the system of ev en ts Ψ consisting of all the eve n ts Q i ( e ) , i = 0 , 1 , e ∈ E , the ev en ts Q 2 ( e, F ) , e ∈ E , F ∈ CONF1( e ) , and also the ev en ts Q 3 ( e, F ) , e ∈ E , F ∈ CONF2( e ) . By (61) the inequalit y holds P ( F ) ≤ P [ B∈ Ψ B ! = 1 − P \ B∈ Ψ B ! . (63) W e shall sho w that the pro ba bility of T B∈ Ψ B is greater than zero. Due to Lo cal Lem ma (see Theorem 6), it is sufficien t to find, for ev ery B ∈ Ψ , a system of ev en ts Ψ B ⊂ Ψ suc h that B and the algebra g enerated by {Q ∈ Ψ \ Ψ B } a re independen t, and, moreov er, suc h that the follo wing inequalit y holds: X Q∈ Ψ B P ( Q ) ≤ 1 / 4 . (64) The ev en t B ∈ Ψ can hav e one of the follow ing three t yp es: 1. B = Q i ( e ) for some e ∈ E and i ∈ { 0 , 1 } ; 2. B = Q 2 ( e, F ) for some e ∈ E and F ∈ CONF1( e ) ; 3. B = Q 3 ( e, F ) for some e ∈ E and F ∈ CONF2( e ) . F or any B ∈ Ψ , w e d efine the domain D ( B ) of the ev en t B as follows: D ( B ) =      e, if B = Q i ( e ) , i = 0 , 1; e ∪ S f ∈ F f ! , if B = Q i ( e, F ) , i = 2 , 3 . 23 By the definitions (38), (41), (45), (53) t he ev en t B b elongs to the algebra generated b y the random v ariables { ξ j , η j : j ∈ D ( B ) } . Then this ev en t is indep enden t of the algebra generated b y the random v ariables { ξ j , η j : j ∈ V \ D ( B ) } . W e tak e the sy stem Ψ B consisting of all the ev en ts Q ∈ Ψ s uc h that the domains of Q and B ha v e no nempty in tersection. In other words, Ψ B = {Q ∈ Ψ : D ( Q ) ∩ D ( B ) 6 = ∅} . Th us, the ev en t B is indep enden t of the algebra generated by {J ∈ Ψ \ Ψ B } , if w e choose Ψ B in this w a y . W e ha v e to ch ec k the inequalit y (64). By the c hoice of the set Ψ B w e get the relation X J ∈ Ψ B P ( J ) ≤ X e ∈ E : e ∩ D ( B ) 6 = ∅ ( P ( Q 0 ( e )) + P ( Q 1 ( e ))) + X e ∈ E , F ∈ CONF1( e ): D ( B ) ∩ D ( Q 2 ( e,F )) 6 = ∅ P ( Q 2 ( e, F )) + + X e ∈ E , F ∈ CONF2( e ): D ( B ) ∩ D ( Q 3 ( e,F )) 6 = ∅ P ( Q 3 ( e, F )) . (65) Let us denote b y a ( B ) , b ( B ) and c ( B ) the num b er of summands in the first sum, the second sum and the third sum in the rig h t-hand side of (65) resp ectiv ely . Using these denotations from the relation (65) and the estim ates (39 ), (4 2), (47), (60) we get the inequalit y X J ∈ Ψ B P ( B ) ≤ a ( B )  r ( r − 1)  p r  k + r 1 − k (1 − q ) k − t − ω ( k q ) t + ω  + + b ( B ) r − ( t +1)( k − 1) q t ( k q ) t ( t + ω − 2) + c ( B ) r − t ( k − 1) (1 + q ) k (2 q ) t − 1 . (66) No w w e s hall consider three case s de p ending on B . 1. B = Q i ( e ) for some e ∈ E and i ∈ { 0 , 1 } . By the condition (30) of Theorem 5 there exis t at most d other edges in tersecting an arbitrary e ∈ E . So, a ( B ) ≤ d + 1 , b ( B ) ≤ ( d + 1)  d t  + ( d + 1) d  d − 1 t − 1  , c ( B ) ≤ ( d + 1)  d t − 1  + ( d + 1) d  d − 1 t − 2  . (67) The first ineq ualit y in (67) is ob vious. T o sho w the last t w o it is sufficien t to notic e that e can inters ect e ither wi th e ′ from the ev en t Q 2 ( e ′ , F ) or with some f ∈ F . 2. B = Q 2 ( e, F ) for some e ∈ E a nd F ∈ CONF1( e ) . T his ev ent dep ends on ( t + 1) edges. So, b y using the estimates from (67) w e get a ( B ) ≤ ( t + 1)( d + 1 ) , b ( B ) ≤ ( t + 1)  ( d + 1)  d t  + ( d + 1) d  d − 1 t − 1  , c ( B ) ≤ ( t + 1)  ( d + 1)  d t − 1  + ( d + 1) d  d − 1 t − 2  . (68) 24 3. B = Q 3 ( e, F ) for some e ∈ E a nd F ∈ CONF2( e ) . This ev en t dep ends on t edges. So, as in the previous c ase a ( B ) ≤ t ( d + 1) , b ( B ) ≤ t  ( d + 1)  d t  + ( d + 1) d  d − 1 t − 1  , c ( B ) ≤ t  ( d + 1)  d t − 1  + ( d + 1) d  d − 1 t − 2  . (69) It is easy to see from (67), (6 8) and (69) that the maximal upp er b ounds for a ( B ) , b ( B ) and c ( B ) are in the second case. So, to prov e (64) it is sufficien t to establish (due to (66)) the follo wing inequalit y: W = ( t + 1)( d + 1)  r ( r − 1)  p r  k + r 1 − k (1 − q ) k − t − ω ( k q ) t + ω  + +( t + 1)  ( d + 1)  d t  + ( d + 1) d  d − 1 t − 1  r − ( t +1)( k − 1) q t ( k q ) t ( t + ω − 2) + + ( t + 1)  ( d + 1)  d t − 1  + ( d + 1) d  d − 1 t − 2  r − t ( k − 1) (1 + q ) k (2 q ) t − 1 ≤ 1 / 4 (70) W e s hall need some additional estim ates con tained in the next section. 3.8 Auxiliary analytics The v alue W (see (7 0)) consists of four summands: ( t + 1)( d + 1) r ( r − 1) ( p/r ) k , ( t + 1)( d + 1) r 1 − k (1 − q ) k − t − ω ( k q ) t + ω , ( t + 1)  ( d + 1)  d t  + ( d + 1) d  d − 1 t − 1  q t r − ( k − 1)( t +1) ( k q ) t ( t + ω − 2) , ( t + 1)  ( d + 1)  d t − 1  + ( d + 1) d  d − 1 t − 2  r − ( k − 1) t (1 + q ) k (2 q ) t − 1 . Consider and estimate them separately . 1. The first summand is ( t + 1)( d + 1) r ( r − 1) ( p/r ) k . Using the restriction (30), the conditions (31) and (32) w e o btain the upp er bound fo r the first summand: ( t + 1)( d + 1) r ( r − 1 )  p r  n ≤ ( t + 1) k r k − 1 r 1 − n ( r − 1)  q r − 1  k = = ( t + 1) k ( r − 1) 1 − k q k ≤ k 2 q k ≤ k 2 2 − k . (71) 2. The second summand is ( t + 1)( d + 1 ) r 1 − k (1 − q ) k − t − ω ( k q ) t + ω . Since the choic e of para meter q in (29), w e g et the relations ( t + 1)( d + 1) r 1 − k (1 − q ) k − t − ω ( k q ) t + ω ≤ ( t + 1 ) k r k − 1 r 1 − k (1 − q ) k − t − ω ( k q ) t + ω = = ( t + 1) n (1 − q ) k − t − ω ( α ln k ) t + ω ≤ ( t + 1) k e q ( t + ω ) − q k ( α ln k ) t + ω = = ( t + 1) k 1 − α e α (ln k )( t + ω ) /k ( α ln k ) t + ω . (72) 25 3. Let u s consider the third summand in the expression (70 ) for the v alue W : ( t + 1)  ( d + 1)  d t  + ( d + 1) d  d − 1 t − 1  q t r − ( k − 1)( t +1) ( k q ) t ( t + ω − 2) . (73) W e s hall need some prelimin ary es timates. First, the following inequalities hold: ( t + 1)  ( d + 1)  d t  + ( d + 1) d  d − 1 t − 1  = ( t + 1)  d t  ( d + 1)( t + 1) ≤ ≤ ( t + 1 ) 2 ( d + 1) d t t ! ≤ ( t + 1 ) 2 ( d + 1) t t ! . (74) Second, the choic e of parameters t and q (see (2 9)) implies the relations q t ( k q ) t ( t + ω − 2) = k − t ( k q ) t ( t + ω − 1) ≤ k − t ( k q ) t 2 + tω = k − t exp  t 2 ln ( α ln k )  ( k q ) tω ≤ ≤ k − t exp { ln k } ( α ln k ) tω = k 1 − t ( α ln k ) tω . (75) Finally , from (74), (75) and the original restriction (30), w e obtain the upper b ound for the expression (73) : ( t + 1)  ( d + 1)  d t  + ( d + 1) d  d − 1 t − 1  q t r − ( k − 1)( t +1) ( k q ) t ( t + ω − 2) ≤ ≤ ( t + 1) 2 t ! ( d + 1) t +1 r − ( k − 1)( t +1) k 1 − t ( α ln k ) tω ≤ ( t + 1) 2 t ! k ( t +1)(1 − b/t ) k 1 − t ( α ln k ) tω ≤ ≤ ( t + 1) 2 t ! k t +1 − ( b ( t +1) /t ) k 1 − t ( α ln k ) tω = = ( t + 1) 2 t ! k 2 − b − ( b/t ) ( α ln k ) tω ≤ ( t + 1) 2 t ! k 2 − b ( α ln k ) tω . (76) 4. It re mains to estimate the fourth summand in the expression (70 ) for t he v alue W : ( t + 1)  ( d + 1)  d t − 1  + ( d + 1) d  d − 1 t − 2  r − ( k − 1) t (1 + q ) k (2 q ) t − 1 . (77) By an analo g y with (74), w e g et: ( t + 1)  ( d + 1)  d t − 1  + ( d + 1) d  d − 1 t − 2  = ( t + 1)  d t − 1  ( d + 1) t ≤ ≤ ( t + 1 ) t  de t − 1  t − 1 ( d + 1) ≤ ( d + 1) t  e t − 1  t − 1 ( t + 1) t. (78) F urther, b y (29) it holds t ha t (2 q ) t − 1 (1 + q ) k ≤ k 1 − t (2 α ln k ) t − 1 e q k = k 1+ α − t (2 α ln k ) t − 1 . (79) 26 Finally , from (78), (79) and (30) we obtain an upp er b ound for the expres sion (77): ( t + 1)  ( d + 1)  d t − 1  + ( d + 1) d  d − 1 t − 2  r − ( k − 1) t (1 + q ) k (2 q ) t − 1 ≤ ≤ ( t + 1) t  e t − 1  t − 1 ( d + 1) t r − ( k − 1) t k 1+ α − t (2 α ln k ) t − 1 ≤ ≤ ( t + 1) t  2 eα ln k t − 1  t − 1 r ( k − 1) t k t (1 − b/t ) r − ( k − 1) t k 1+ α − t = = ( t + 1) t  2 eα ln k t − 1  t − 1 k 1+ α − b . (80) The inequalit y (80 ) completes the estimation of the parts of the v alue W . 3.9 The completion of the pro of of Theorem 5 Let us gather the obtained b ounds for the summands in the expression (70) for the v alue W . The relations (71), (72), (7 6) and (80) imply the ine qualities W ≤ k 2 2 k + ( t + 1) k 1 − α e α (ln k )( t + ω ) /k ( α ln k ) t + ω + ( t + 1) 2 t ! k 2 − b ( α ln k ) tω + +( t + 1) t  2 eα ln k t − 1  t − 1 k 1+ α − b < 1 4 , the last of whic h holds, since the condi tion ( 3 3) of theorem 5. Th us, the required rel ation (70) is establish ed. It implies the inequalit y (64) necessary for the application o f Lo cal Lemma. It follo ws from Lo cal Lemma that the probabilit y of sim ultaneous happening of all the ev en t s B , where B ∈ Ψ , is greater than zero. Then b y (6 3) w e hav e sho wn that P ( F ) < 1 . Let us complete the proof. Indeed, w e ha v e prov ed, that the probabilit y of the ev ent that the random coloring ~ ζ is not a prop er coloring of H is less than one. So, ~ ζ is a proper coloring with p ositiv e probabilit y , and χ ( H ) ≤ r . Theorem 5 is pro ved. 3.10 The completion o f the pro of of Theorem 4 W e s hall use Theorem 5. Let us c ho ose the parameters b and α : b = 4 , α = 2 . By this c hoice of b , α and the condition ω ≤ p ln k / (ln ln k ) there exis ts an inte ger k 1 suc h that for all k ≥ k 1 , the inequalities (31) and (32) hold. Let us consider the left part of (33 ). W e hav e t = O  p ln k / ln ln k  (see (29)), so ( t + 1) k 1 − α e α (ln k )( t + ω ) /k ( α ln k ) t + ω = e O (ln ln k ) k − 1 e o (1) e O ( √ ln k ln ln k ) = o (1) , k → ∞ , 27 ( t + 1) 2 t ! k 2 − b ( α ln k ) tω = O  k − 2  e ln k (1+ o (1)) = o (1) , k → ∞ , ( t + 1) t  2 eα ln k t − 1  t − 1 k 1+ α − b = e O ( √ ln k ln ln k ) k − 1 = o (1) , k → ∞ . These relations imply the existence o f an in teger k 2 suc h that the inequalit y (33 ) holds, for all k ≥ k 2 . Let H = ( V , E ) b e an k -uniform h yp ergraph, H ∈ H ( k , r, ω ) with ω ≤ p ln k / (ln ln k ) . In the case k ≥ k 0 = max( k 1 , k 2 ) the hypergraph H satisfies all the conditions of Theorem 5, except (30). But H is not r -colorable, and so there exists an edge e ∈ E with edge degree at least  r k − 1 k 1 − b/t  . So, the edge e con tains a v ertex with degree a t least  r k − 1 k 1 − b/t  /k + 1 ≥  r k − 1 k 1 − b/t − 1  /k + 1 = r k − 1 k − b/t + 1 − 1 /k . Th us, w e hav e establis hed the inequalit y ∆( H ) > r k − 1 k − b/t and, consequen tly , ∆( H ( n, r, ω )) ≥ r k − 1 k − b/t = r k − 1 k − 4 j q ln k ln(2 ln k ) k − 1 . Theorem 4 is prov ed. 4 Cho osabilit y in random h yp ergraphs In this section we will discuss r -choosability of the random hy p ergraph H ( n, k , p ) . Let us recall the required definitions. W e sa y that a h ypergraph H is r - ch o osab le if for ev ery family of sets L = { L ( v ) : v ∈ V } ( L is called list assign m ent ), suc h that | L ( v ) | = r for all v ∈ V , there is a prop er coloring f r o m the lists (f o r ev ery v ∈ V we should use a color from L ( v ) ). The choic e n umb er of a hy p ergraph H , denoted by ch ( H ) , is the least r suc h that H is r -choosable. It is clear that χ ( H ) ≤ ch ( H ) . The choic e n um b ers of graphs we re indep enden tly in tro duced b y V.G . Vizing (see [20]) and b y P . Erdos, A. Rubin and H. T aylor (see [21]). In this pap er w e consider the threshold pro bability for r -choosability of H ( n, k , p ) . 4.1 Threshold for r -c ho osabi lit y in H ( n, k , p ) The choice num b er of the random h yp ergraph H ( n, k , p ) has b een studied b y M. Kriv elevic h and V. V u (see [22]). They prov ed that ch ( H ( n, k , p )) is asymptotically v ery closed to χ ( H ( n, k , p )) . Their first resul t holds f o r almost all p , but has at little gap betw een ch ( H ( n, k , p )) and χ ( H ( n, k , p )) . Theorem 7. (M. Kriv elevic h, V. V u, [22]) Supp ose k ≥ 2 is fixe d. Th er e exists a c onstant C = C ( k ) such t hat, for any p satisfying C n 1 − k ≤ p ≤ 0 . 9 , the fo l lowing c onver genc e holds P  ch ( H ( n, k , p )) ≤ (1 + ψ ( n )) k 1 / ( k − 1) χ ( H ( n, k , p ))  → 1 as n → ∞ , wher e ψ ( n ) → 0 as n k − 1 p → ∞ . The second theorem from [22] states that, for sufficien tly large p , this gap can b e remov ed. 28 Theorem 8. (M. Kriv elevi c h, V. V u, [22]) Supp ose k ≥ 2 is fixe d and 0 < ε < ( k − 1) 2 / (2 k ) . Then, for any p satisfying n − ( k − 1) 2 / (2 k )+ ε ≤ p ≤ 0 . 9 , the fol low ing c onver ge nc e holds P ( ch ( H ( n, k , p )) = (1 + o (1)) χ ( H ( n, k , p ))) → 1 as n → ∞ . Theorem 7 together w ith Theorem 2 of Krivele vic h and Sudak ov impli es the f o llo wing corollary , whic h is an analogue of Corollary 1 f or r -choosability . Corollary 3. L et k ≥ 3 and ε ∈ (0 , 1) b e fi xe d. Ther e is a c onstant r 0 = r 0 ( k , ε ) such that, for any r = r ( n ) satisfying the c onditions r ≥ r 0 , r k − 1 ln r = o ( n k − 1 ) , the fol lowing c on v e r genc e holds : P ( ch ( H ( n, k , p )) ≤ r ) → 1 , wher e p = (1 − ε ) r k − 1 ln r k n  n k  . W e see that the low er b ound for the threshold for r -c ho osabilit y pro vided b y Corollary 3 do es not coincide with the upp er b ound in Lemma 2 (if hypergraph is not r -colorable then it is also not r -ch o osable). Thei r ratio has an order of k . Recall th at applying Theorem 2 pro vid es the follow ing restrictions on the parameters r and k in Corollary 3 (see (9)): r = Ω  k 29 (ln k ) 28  , n ≥ k 9 k + O ( k ln ln k / (ln k )) . (81) What can b e said ab out the low er b o und for the threshold for r -ch o osability when r = O ( k 29 (ln k ) 28 ) ? R ema rk 3 . It should b e noted that, for v ery large r (e.g., r > √ n ) and fixed k , Theorem 8 together with Theorem 2 giv es a n asymptotic v alue fo r the required threshold for r -c ho osability : p ∗ ∼ r k − 1 ln r n  n k  . The pro of of Theorem 1 by Ac hlioptas, Kriv elevic h, Kim and T etali is based on the determi- nistic coloring algorithm, whic h cannot be generalized to the case of an arbitrary r -uniform list assignmen t, so the low er b ound (7) do es not hold for the threshold probabilit y for r -choosability . Moreo v er, t he pro of of the result (4) b y A c hlioptas a nd Mo ore is also cannot b e adopted for list colorings. Th us, in the case when r is small in c omparison with k w e ha v e only the result of Lemma 1 whic h is j ust a generalization of the res ult (1) b y Alon and Sp encer. Lemma 4. Ther e exists an inte ger k 0 and a p ositive numb er c such that for any fixe d k ≥ k 0 and r ≥ 2 , the fol lowing statement h olds: if p ≤ c r k − 1 k 2 n  n k  , then P ( H ( n, k , p ) is r -cho o sable ) → 1 . (82) Our new results concerning r -choosability in random h ypergraphs a re form ulated in the follo wing t w o the orems. 29 Theorem 9. S upp ose k = k ( n ) ≥ 3 and r = r ( n ) ≥ 3 satisfy the r elation 3 128 r k − 1 √ k − ln n → −∞ as n → ∞ . If p ≤ 3 32 r k − 1 k 3 / 2 n  n k  , (83) then P ( H ( n, k , p ) is r -cho osable ) → 1 . Theorem 10. Supp ose δ ∈ (0 , 1) is a c onstant. L et k = k ( n ) and r = r ( n ) ≥ 2 satisfy the fol lowing c onditions: k ≥ k 0 , wher e k 0 is some absolute c onstant, and, mor e over, ( k − 1) ln r < 1 − δ 2 ln n, r k − 1 k − ϕ ( k ) ≥ 6 ln n, wher e ϕ ( k ) = 4 j q ln k ln(2 ln k ) k − 1 . Then f o r function p = p ( n ) , satisfying p ≤ 1 2 r k − 1 k 1+ ϕ ( k ) n  n k  , (84) we have P ( H ( n, k , p ) is r -cho osable ) → 1 as n → ∞ . It is easy to see that Theorem 9 and Theorem 1 0 stated that the results of Corolla ry 2 (assertion 1)) a nd Theorem 3 also hold in the case of list colorings. Both pro vided b ounds are b etter (fo r all sufficien tly large k ) than the result (82) of Lemma 4, and b oth are w orse than the result of Corollary 3. Hence, Theorem 10 giv es the b est lo w er b ound (18) f or the threshold probabilit y for r - choosability of H ( n, k , p ) in the wide area of the parameters (recall the restriction (81)): r ≤ k 29 (ln k ) 28 and 6 k ϕ ( k ) ln n ≤ r k − 1 ≤ n (1 − δ ) / 2 , where k ≥ k 0 is sufficien tly large. The ine qualit y (17), in c omparison with (18), does not ha v e an upp er restriction r k − 1 ≤ n (1 − δ ) / 4 , so it prov ides the best lo wer bo und in the a rea 3 ≤ r ≤ k 29 (ln k ) 28 and r k − 1 ≥ n (1 − δ ) / 2 . The pro of s of Theorems 9 and Theorem 10 are v ery similar to the pro ofs of the first assertion of Corolla ry 2 and Theorem 3, so w e do not giv e the complete arg umen t and describ e only the main ides and differences. 4.2 Ideas of the pro ofs of Theorems 9 a nd 10 F or giv en k , r ≥ 2 , let ∆ list ( k , r ) denote the minim um p ossible ∆( H ) , where H is a k - uniform non- r -choosable hypergraph. In [1 1] D.A. Shabano v show s that the low er b ound (12) for ∆( k , r ) (see § 2.1) holds fo r ∆ list ( k , r ) a lso: for an y k , r ≥ 3 , ∆ list ( k , r ) > 1 8 k − 1 / 2 r k − 1 . Using this inequalit y one c an easily pro v e The orem 9 by the same argumen t a s in Lemma 3. 30 R ema rk 4 . The low er b ound (13) for ∆( k , r ) obtained by K ostochk a, Kum bhat and R¨ odl do es not hold for ∆ list ( k , r ) , so w e c annot appl y it to r - c ho osabilit y of random h yp ergraphs. T o pro v e Theorem 10 it is sufficien t to sho w that under the conditions of Theorem 5 the h yp ergraph H is not only r -colorable, but is r -choosable. The pro of of r -choosability remains almost the same. The difference appears in the distribu tions of the random v ariables. Supp ose H = ( V , E ) is a k -unifor m hypergraph satisfying the conditions o f Theorem 5 and let L = { L ( v ) : v ∈ V } b e an r -uniform list a ssignmen t with the set of colors N . Without loss o f g enerality , V = { 1 , . . . , w } . In comparison with § 3.3 w e introduce ra ndom v ariables with another distribution. Let ξ 1 , . . . , ξ w and η 1 , . . . , η w , b e m utually indep enden t ra ndom v ariables with the followin g distribution: • ξ i , i = 1 , . . . , w , has the u niform distribution o n the set L ( i ) ( i = 1 , . . . , w ), • η i , i = 1 , . . . , w , tak es a ll v alues from L ( i ) with the same probabilit y p and the v alue 0 with probability 1 − r p . 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