YOSHIHARU KOHAYAKAWA, 1 VOJTĚCH RÖDL,2 and MATHIAS SCHACHT 2

Transcription

1 Combnatorcs, Probablty and Computng 19XX 00, c 19XX Cambrdge Unversty Press Prnted n the Unted Kngdom The Turán Theorem for Random Graphs YOSHIHARU KOHAYAKAWA, 1 VOJTĚCH RÖDL, and MATHIAS SCHACHT 1 Insttuto de Matemátca e Estatístca,Unversdade de São Paulo, Rua do Matão 1010, São Paulo, Brazl e-mal: yosh@me.usp.br Department of Mathematcs and Computer Scence, Emory Unversty, Atlanta, GA 303, USA e-mal: {rodl, mschach}@mathcs.emory.edu The am of ths paper s to prove a Turán type theorem for random graphs. For 0 < γ and graphs G and H, wrte G γ H f any γ-proporton of the edges of G spans at least one copy of H n G. We show that for every graph H and every fxed real δ > 0 almost every graph G n the bnomal random graph model Gn, q, wth q = qn log n 4 /n 1/dH, satsfes G χh /χh 1+δ H, where as usual χh denotes the chromatc number of H and dh s the degeneracy number of H. Snce K l, the complete graph on l vertces, s l-chromatc and l 1-degenerate we nfer that for every l and every fxed real δ > 0 almost every graph G n the bnomal random graph model Gn, q, wth q = qn log n 4 /n 1/l 1, satsfes G l /l 1+δ K l. CONTENTS 1 Introducton Prelmnary results 4.1 Prelmnary defntons 4. The regularty lemma for sparse graphs 4.3 The countng lemma for complete subgraphs of random graphs 5 3 The man result Propertes of almost all graphs 6 3. A determnstc subgraph lemma Proof of the man result 10 Research supported n part by MCT/CNPq through ProNEx Programme Proc. CNPq /1997 4, by CNPq Proc /93 1 and / Research supported n part by an NSF Grant The collaboraton of the authors s supported by a CNPq/NSF cooperatve grant /99 7,

2 Y. Kohayaawa, V. Rödl, and M. Schacht 4 The countng lemma The pc-up lemma The -tuple lemma for subgraphs of random graphs Outlne of the proof of the countng lemma for l = Proof of the countng lemma 1 5 The d-degenerate case 8 References Introducton A classcal area of extremal graph theory nvestgates numercal and structural problems concernng H-free graphs, namely graphs that do not contan a copy of a gven fxed graph H as a subgraph. Let exn, H be the maxmal number of edges that an H-free graph on n vertces may have. A basc queston s then to determne or estmate exn, H for any gven H and large n. A soluton to ths problem s gven by the celebrated Erdős Stone Smonovts theorem, whch states that, as n, we have exn, H = 1 1 χh 1 + o1 n, 1 where as usual χh s the chromatc number of H. Furthermore, as proved ndependently by Erdős and Smonovts, every H-free graph G = G n that has as many edges as n 1 s n fact very close n a certan precse sense to the densest n-vertex χh 1-partte graph. For these and related results, see, for nstance, Bollobás [1]. Here we are nterested n a varant of the functon exn, H. Let G and H be graphs, and wrte exg, H for the maxmal number of edges that an H-free subgraph of G may have. Formally, exg, H = max{ EF : H F G}. For nstance, f G = K n, the complete graph on n vertces, then exk n, H = exn, H s the usual Turán number of H. Our am here s to study exg, H when G s a random graph. Let 0 < q = qn 1 be gven. The bnomal random graph G n Gn, q has as ts vertex set a fxed set V G of cardnalty n and two vertces are adjacent n G wth probablty q. All such adjacences are ndependent. For concepts and results concernng random graphs not gven n detal below, see, e.g., Bollobás []. Here we wsh to nvestgate the random varables exgn, q, H, where H = K l l or H s a d-degenerate graph, a graph that may be reduced to the empty graph by the successve removal of vertces of degree less or equal d. Let H be a graph of order H = V H 3. Let us wrte d H for the -densty of H, that s, { eh } 1 d H = max H : H H, H 3. A general conjecture concernng exgn, q, H, frst stated n [10], s as follows as s usual n the theory of random graphs, we say that a property P holds almost surely

3 The Turán Theorem for Random Graphs 3 or that almost every random graph G n Gn, q satsfes P f P holds wth probablty tendng to 1 as n. Conjecture 1.1. Let H be a non-empty graph of order at least 3, and let 0 < q = qn 1 be such that qn 1/dH as n. Then almost every G n Gn, q satsfes exg, H = 1 1 χh 1 + o1 EG. In other words, for G n Gn, q the Conjecture 1.1 clams that G γ H holds almost surely for any fxed γ > 1 1/χH 1. There are a few results n support of Conjecture 1.1. Any result concernng the tree-unversalty of expandng graphs, or any smple applcaton of Szemeréd s regularty lemma for sparse graphs see Theorem. below, gves Conjecture 1.1 for H a forest. The cases n whch H = K 3 and H = C 4 are essentally proved n Franl and Rödl [3] and Füred [4], respectvely, n connecton wth problems concernng the exstence of some graphs wth certan extremal propertes. The case for H = K 4 was proved by Kohayaawa, Lucza, and Rödl [10] and the case n whch H s a general cycle was settled by Haxell, Kohayaawa, and Lucza [5, 6] see also Kohayaawa, Kreuter, and Steger [9]. Our man result relates to Conjecture 1.1 n the followng way: we deal wth the case n whch H = K l and q = qn log n 4 /n 1/l 1. More precsely we prove the followng. Theorem 1.. Let l, q = qn log n 4 /n 1/l 1, and let Gn, q be the bnomal random graph model wth edge probablty q. Then for every 1/l 1 > δ > 0 a graph G n Gn, q satsfes the followng property wth probablty 1 o1: If F s an arbtrary, not necessarly nduced subgraph of G wth EF 1 1 n l 1 + δ q, then F contans K l, the complete graph on l vertces, as a subgraph. Moreover, there exsts a constant c = cδ, l such that F contans at least cq l n l copes of K l. In ths paper we gve a proof of Theorem 1.. In Secton 5 we outlne the proof of an extenson of ths result, Theorem 1. the detaled proof s gven n [14]. Recall that a graph H wth V H = h s d-degenerate f there exsts an orderng of the vertces v 1,..., v h such that each v 1 h has at most d neghbours n Very recently, Szabó and Vu [16] proved ndependently the same result under a slghtly weaer assumpton; n fact, they proved Theorem 1. for qn n 1/l 3/. Ther proof s elegant. To obtan the smaller lower bound for q, they mae use of the fact that Conjecture 1.1 holds for H = K 4 [10] as the base of an nducton; wthout usng ths result, ther proof gves essentally the same condton on q as ours. Ther approach extends to several nfnte famles of graphs H see [16, Secton 4]; the present proof extends to all graphs, and wors for qn log n 4 /n 1/d, where d = dh s the degeneracy number of the graph H; see Theorem 1..

4 4 Y. Kohayaawa, V. Rödl, and M. Schacht {v 1,..., v 1 } for more detals concernng d-degenerate graphs see [13, 15]. Snce K l s clearly l 1-degenerate and l-chromatc, the followng result extends Theorem 1.. Theorem 1.. Let d be a postve nteger, H a d-degenerate graph on h vertces, q = qn log n 4 /n 1/d, and Gn, q the bnomal random graph model wth edge probablty q. Then for every 1/χH 1 > δ > 0 a graph G n Gn, q satsfes the followng property wth probablty 1 o1: If F s an arbtrary, not necessarly nduced subgraph of G wth 1 n EF 1 χh 1 + δ q, then F contans H as a subgraph. Moreover, there exsts a constant c = cδ, H such that F contans at least cq EH n h copes of H. Ths paper s organzed as follows. In Secton we descrbe a sparse verson of Szemeréd s regularty lemma Theorem. and we state the countng lemma Lemma.3, whch are crucal n our proof of Theorem 1.. We prove Theorem 1. n Secton 3. Secton 4 s entrely devoted to the proof of Lemma.3. The proof of Lemma.3 reles on the Pc-Up Lemma Lemma 4.3 and on the -tuple lemma Lemma 4.7. We gve these prelmnary results n Secton In Secton 4.3 we outlne the proof of Lemma.3 n the case l = 4. Fnally, the proof s gven n Secton 4.4. We dscuss the case when H s a d-degenerate graph and setch the proof of Theorem 1. n Secton 5. For a general remar about the notaton we use throughout ths paper see the remar n Secton.3. Acnowledgement. The authors than the referee for hs or her detaled wor.. Prelmnary results.1. Prelmnary defntons Let a graph G = G n of order V G = n be fxed. For U, W V = V G, we wrte { } EU, W = E G U, W = {u, w} EG: u U, w W for the set of edges of G that have one end-vertex n U and the other n W. Notce that each edge n U W occurs only once n EU, W. We set eu, W = e G U, W = EU, W. If G s a graph and V 1,..., V t V G are dsjont sets of vertces, we wrte G[V 1,..., V t ] for the t-partte graph naturally nduced by V 1,..., V t... The regularty lemma for sparse graphs Our am n ths secton s to state a varant of the regularty lemma of Szemeréd [17]. Let a graph H = H n = V, E of order V = n be fxed. Suppose ξ > 0, C > 1, and 0 < q 1. Defnton.1 ξ, C-bounded. For ξ > 0 and C > 1 we say that H = V, E s

5 The Turán Theorem for Random Graphs 5 a ξ, C-bounded graph wth respect to densty q, f for all U, W V, not necessarly dsjont, wth U, W ξ V, we have U W e H U, W Cq U W. For any two dsjont non-empty sets U, W V, let d H,q U, W = e HU, W q U W. We refer to d H,q U, W as the q-densty of the par U, W n H. When there s no danger of confuson, we drop H from the subscrpt and wrte d q U, W. Now suppose ε > 0, U, W V, and U W =. We say that the par U, W s ε, H, q-regular, or smply ε, q-regular, f for all U U, W W wth U ε U and W ε W we have d H,q U, W d H,q U, W ε. 3 Below, we shall sometmes use the expresson ε-regular wth respect to densty q to mean that U, W s an ε, q-regular par. We say that a partton P = V t 0 of V = V H s ε, t-equtable f V 0 εn, and V 1 = = V t. Also, we say that V 0 s the exceptonal class of P. When the value of ε s not relevant, we refer to an ε, t-equtable partton as a t-equtable partton. Smlarly, P s an equtable partton of V f t s a t-equtable partton for some t. We say that an ε, t-equtable partton P = V t 0 of V s ε, H, q-regular, or smply ε, q-regular, f at most ε t pars V, V j wth 1 < j t are not ε, q-regular. We may now state a verson of Szemeréd s regularty lemma for ξ, C-bounded graphs. Theorem.. For any gven ε > 0, C > 1, and t 0 1, there exst constants ξ = ξε, C, t 0 and T 0 = T 0 ε, C, t 0 t 0 such that any suffcently large graph H that s ξ, C-bounded wth respect to densty 0 < q 1 admts an ε, H, q-regular ε, t-equtable partton of ts vertex set wth t 0 t T 0. A smple modfcaton of Szemeréd s proof of hs lemma gves Theorem.. For applcatons of ths varant of the regularty lemma and ts proof, see [8, 1]..3. The countng lemma for complete subgraphs of random graphs Let t l be fxed ntegers and n a suffcently large nteger. Let α and ε be constants greater than 0. Let G Gn, q be the bnomal random graph wth edge probablty q = qn, and suppose J s an l-partte subgraph of G wth vertex classes V 1,..., V l. For all 1 < j l we denote by J j the bpartte graph nduced by V and V j. Consder the followng assertons for J. I V = m = n/t II q l 1 n log n 4 III J j has T = pm edges where 1 > αq = p 1/n, and IV J j s ε, q-regular.

6 6 Y. Kohayaawa, V. Rödl, and M. Schacht Remar. Strctly speang, n I we should have, say, m/t, because m s an nteger. However, throughout ths paper we wll omt the floor and celng sgns and, snce they have no sgnfcant effect on the arguments. Moreover, let us mae a few more comments about the notaton that we shall use. For postve functons fn and gn, we wrte fn gn to mean that lm n gn/fn = 0. Unless otherwse stated, we understand by o1 a functon approachng zero as the number of vertces of a gven random graph goes to nfnty. Fnally, we observe that our logarthms are natural logarthms. We are nterested n the number of copes of complete graphs on l vertces n such a subgraph J satsfyng condtons I IV. Lemma.3 Countng lemma. For every α, σ > 0 and nteger l there exsts ε > 0 such that for every fxed nteger t l a random graph G n Gn, q satsfes the followng property wth probablty 1 o1: Every subgraph J G satsfyng condtons I IV contans at least copes of the complete graph K l. 1 σp l m l We wll prove Lemma.3 later n Secton The man result In ths secton we wll prove the man result of ths paper, Theorem 1.. Ths secton s organzed as follows. Frst, we state two propertes that hold for almost every G Gn, q. Then, n Secton 3., we prove a determnstc statement about the regularty of certan subgraphs of an ε, q-regular α-dense t-partte graph. Fnally, we prove Theorem Propertes of almost all graphs We start wth a well nown fact of random graph theory whch follows easly from the propertes of the bnomal dstrbuton. Fact 3.1. If G s a random graph n Gn, q, then n EG = 1 + o1 q holds wth probablty 1 o1. The next property refers to Defnton.1 and wll enable us to apply Theorem.. Lemma 3.. For every C > 1, ξ > 0 and q = qn 1/n a random graph G n Gn, q s ξ, C-bounded wth probablty 1 o1.

7 The Turán Theorem for Random Graphs 7 We wll apply the followng one-sded estmate of a bnomally dstrbuted random varable. Lemma 3.3. Let X be a bnomal dstrbuted random varable n BN, q wth expectaton EX = Nq and let C > 1 be a constant. Then PX CEX exp τcex, where τ = log C 1 + 1/C > 0 for C > 1 recall that all logarthms are to base e, see the remar n Secton.3. Proof. The proof s gven n [7] see Corollary.4. Proof of Lemma 3.. Let G Gn, q and let U, W V G be two not necessarly dsjont sets such that U, W ξn. Clearly, eu, W s a bnomal random varable wth U W E[eU, W ] = q U W. Observe that E[eU, W ] n snce q 1/n. Set τ = log C 1 + 1/C. Then Lemma 3.3 mples P eu, W > CE[eU, W ] exp τce[eu, W ]. We now sum over all choces for U and W to deduce that PG s not ξ, C-bounded U ξn W ξn snce τc > 0 and E[eU, W ] n. n n exp τce[eu, W ] U W 4 n exp τce[eu, W ] = o1, 3.. A determnstc subgraph lemma The next lemma states that every ε, q-regular, bpartte graph wth at least αqm edges contans an 3ε, q-regular subgraph wth exactly αqm edges. Lemma 3.4. For every ε > 0, α > 0, and C > 1 there exsts m 0 such that f H = U, W ; F s a bpartte graph satsfyng U = m 1, W = m > m 0, Cqm 1 m e H U, W αqm 1 m for some functon q = qm 0 1/m 0, and H s ε, q-regular, then there exsts a subgraph H = U, W ; F H such that e H U, W = αqm 1 m and H s 3ε, q-regular.

8 8 Y. Kohayaawa, V. Rödl, and M. Schacht Proof. We select a set D of D = e H U, W αqm 1 m dfferent edges n E H U, W unformly at random and fx H = U, W ; F \ D. We naturally defne the densty n D wth respect to q for sets U U and W W by d D,q U, W = E HU, W D q U W. 4 In order to chec the 3ε, H, q-regularty of U, W, t s enough to verfy the nequalty correspondng to 3 for sets U U, W W such that U = 3εm 1 and W = 3εm. Let U, W be such a par. We dstngush three cases dependng on D and e H U, W. Case 1 D ε 3 qm 1 m. The graph H s ε, H, q-regular and thus d H,q U, W d H,q U, W ε. Snce d H,qU, W d H,q U, W d D,q U, W, we have d H,qU, W d H,q U, W whch mples that H s 3ε, q-regular. D 9ε qm 1 m d H,q U, W 10 9 ε, Case e H U, W ε 3 qm 1 m. Observe that e H U, W ε 3 qm 1 m mples H s ε, H, q-regular and thus d H,q U, W ε 9. 5 d H,q U, W ε + d H,q U, W 10 ε. 6 9 On the other hand, d H,qX, Y d H,q X, Y for arbtrary X U and Y W, whch combned wth 5 and 6 yelds d H,qU, W d H,qU, W 10 9 ε + ε 9 3ε. Up to now, we have not used the fact that D s chosen at random. To deal wth the case that we are left wth that s, the case n whch D > ε 3 qm 1 m and e H U, W > ε 3 qm 1 m, we wll mae use of ths randomness. Before we start, we state the followng two-sded estmate for the hypergeometrc dstrbuton. Lemma 3.5. Let sets B U be fxed. Let U = u and B = b. Suppose we select a d-set D unformly at random from U. Then, for 3/ λ > 0, we have P D B bd u λbd exp λ bd. u 3 u Proof. For the proof we refer to [7] Theorem.10.

9 The Turán Theorem for Random Graphs 9 We contnue wth the proof of Lemma 3.4. Case 3 D > ε 3 qm 1 m and e H U, W > ε 3 qm 1 m. Recall that U U and V V are such that U = 3εm 1 and V = 3εm. Frst, we verfy that d D,qU, W d H,qU, W d H,q U, W d D,qU, W ε 7 mples that d H,qU, W d H,qU, W 3ε. 8 Indeed, straghtforward calculaton usng the ε, q-regularty of H and 7 gve d H,qU, W d H,qU, W = d H,q U, W d D,q U, W d H,q U, W d D,q U, W ε + d D,q U, W d D,q U, W ε + d D,qU, W d D,q U, W d H,qU, W d H,q U, W + d D,qU, W d H,qU, W d H,q U, W d D,qU, W ε + d D,qU, W d H,q U, W d H,qU, W d H,q U, W + ε ε + d D,qU, W d H,q U, W ε + ε 3ε. Next, we wll prove that 7 s unlely to fal, because of the random choce of D. We set { 9ε 3 λ = mn C, 3 }. 9 Then the two-sded estmate n Lemma 3.5 gves that D E HU, W e HU, W D e H U, W < λe HU, W D e H U, W fals wth probablty exp λ 3 e H U, W D. 10 e H U, W Snce d D,qU, W d D,q U, W d H,qU, W d H,q U, W 1 = 9ε qm 1 m D E HU, W e HU, W D e H U, W, and because of and 9, we have λ e HU, W 9qε m 1 m D e H U, W λe HU, W 9qε λ e HU, W m 1 m 9qε ε, m 1 m

10 10 Y. Kohayaawa, V. Rödl, and M. Schacht we nfer that 7 and consequently 8 fals wth small probablty gven n 10. We now sum over all possble choces for U and W and use the condtons of ths case.e. D > ε 3 qm 1 m, e H U, W > ε 3 qm 1 m and. We have that P H s not 3ε, q-regular m1+m exp λ ε 6 3C qm 1m < 1 for m 1, m suffcently large, snce q = qm 0 1/m 0. Ths mples that, for m 0 large enough, there s a set D such that H s 3ε, q-regular, as requred Proof of the man result The proof of Theorem 1. s based on Lemma.3, whch we prove later n Secton 4. The man dea s to fnd a regular subgraph J satsfyng I IV of the Countng Lemma, n the arbtrary subgraph F wth EF 1 1 n l 1 + δ q. Proof of Theorem 1.. Let l and 1/l 1 > δ > 0 be fxed and suppose q = qn log n 4 /n 1/l 1. Frst we defne some constants that wll be used n the proof. We start by settng α = δ 8, 11 σ = As a matter of fact, our proof s not senstve to the value of the constant σ; n fact, as long as 0 < σ < 1, every choce wors. We want to use the Countng Lemma, Lemma.3, n order to determne the value of ε. Set α CL = α and σ CL = σ, then Lemma.3 yelds ε CL. We set { ε CL ε = mn 3, δ } and C = 4 + δ We then apply the sparse regularty lemma Theorem. wth ε SRL = ε, C SRL = C and t SRL 0 = max{ 8l /δ, 40/δ}. Theorem. then gves ξ SRL and we defne ξ = ξ SRL. Moreover, Theorem. yelds { } 8l T0 SRL t = t SRL t SRL 0 = max δ, δ For the rest of the proof all the constants defned above α, σ, ε, C, ξ, and t are fxed.

11 The Turán Theorem for Random Graphs 11 Fact 3.1, Lemma 3., and Lemma.3 mply that a graph G n Gn, q satsfes the followng propertes P1 P3 wth probablty 1 o1: P1 EG 1 + o1 q n, P G s ξ, C-bounded, and P3 G satsfes the property consdered n Lemma.3. We wll show that f a graph G satsfes P1 P3, then any F G wth EF 1 1/l 1 + δq n contans at least cq l n l for some constant c = cδ, l copes of K l, and Theorem 1. wll follow. To acheve ths, we frst regularse F by applyng Theorem. wth ε SRL = ε, C SRL = C and t SRL 0 = max{ 8l /δ, 40/δ}. Consequently F admts an ε, q-regular ε, t-equtable partton V t 0. We set m = n/t = V for 0. Let F cluster be the cluster graph of F wth respect to V t 0 defned as follows V F cluster = {1,..., t}, { E F cluster = {, j}: V, V j s ε, q-regular e F V, V j αqm }. Our next am s to apply the classcal Turán theorem to guarantee the exstence of a K l F cluster. For ths we defne a subgraph F of F. Set EF = {E F V, V j : {, j} EF cluster } We now want to fnd a lower bound for EF. There are four possble reasons for an edge e EF not to be n EF : R1 e has at least one vertex n V 0, R e s contaned n some vertex class V for 1 t, R3 e s n EV, V j for an ε, q-rregular par V, V j, or R4 e s n EV, V j for sparse a par.e., ev, V j < αqm. We bound the number of dscarded edges of type R1 R3 by applyng that G s ξ, C- bounded Property P: # of edges of type R1 Cqεn, n # of edges of type R Cq t, t n t # of edges of type R3 Cq ε. t Furthermore, we bound the number of dscarded edges of type R4, by n # of edges of type R4 αq t t.

12 1 Y. Kohayaawa, V. Rödl, and M. Schacht Ths, combned wth n, 11, 13, 14, 15, and δ < 1 mples that EF \ EF C ε + 1 t + ε + α qn C ε α n 4q t δ 4 + δ 40 + δ + δ n q δ n 40 4 q, and thus EF 1 1 l 1 + δ n q. We use the last nequalty and once agan P to acheve the desred lower bound for EF cluster. Indeed, EF cluster ef Cqn/t = 1 1 l 1 + δ δ 1 t n 4, and then, for n large enough n > 16/δ, by usng t 8l /δ, we deduce that EF cluster > 1 1 l 1 + δ 1 δ t l 1 + δ t t l 1 + l. The last nequalty mples, by Turán s theorem [18], that there s a subgraph K l n F cluster. Let { 1,..., l } be the vertex set of ths K l n F cluster. Then we set J 0 = F [V 1,..., V l ] F. Now, every par V j, V j for 1 j < j l satsfes the condtons of Lemma 3.4 wth ε Lem3.4 = ε and α Lem3.4 = α. Thus there s a subgraph J J 0 F that s 3ε, q-regular and e J V j, V j = αqm. Snce ε ε CL /3 and J satsfes condtons I IV of the Countng Lemma, Lemma.3, wth the constants chosen above α CL = α, σ CL = σ, and ε CL 3ε, there are at least 1 σp l m l = 1 σαl q l n l t l l 1 σα T SRL 0 l q l n l dfferent copes of K l n J F. Observe that α, σ and T 0 depend on δ and l but not on n. Consequently, there are cδ, lq l n l 1 where cδ, l = 1 σα l / T SRL l 0 copes of K l n F, as requred by Theorem The countng lemma Our am n ths secton s to prove Lemma.3. In order to do ths, we wll need two lemmas. We ntroduce these n the frst two subsectons. Then, n Secton 4.3, we wll llustrate the proof of the Countng lemma on the partcular case l = 4. Fnally, we gve the proof of Lemma.3 n Secton 4.4.

13 The Turán Theorem for Random Graphs The pc-up lemma Before we state the Pc-Up Lemma, Lemma 4.3, let us state a smple one-sded estmate for the hypergeometrc dstrbuton, whch wll be useful n the proof of Lemma 4.3. Lemma 4.1 A hypergeometrc tal lemma. Let b, d, and u be postve ntegers and suppose we select a d-set D unformly at random from a set U of cardnalty u. Suppose also that we are gven a fxed b-set B U. Then we have for λ > 0 P D B λ bd e λbd/u. 17 u λ Proof. For the proof we refer the reader to [11]. We now state and prove the Pc-Up Lemma. Let be a fxed nteger and let m be suffcently large. Let V 1,..., V be parwse dsjont sets all of sze m and let B be a subset of V 1 V. For 1 > p = pm 1/m set T = pm and consder the probablty space V1 V V 1 V Ω =, T T denotes the famly of all subsets of V V of sze T, and all the R = where V V T R 1,..., R 1 Ω are equprobable,.e., have probablty m 1. T For every R = R 1,..., R 1 Ω the degree wth respect to R 1 < of a vertex v n V s d R v = {v V : v, v R }. 18 Defnton 4. Πζ, µ, K. For ζ, µ, K wth 1 > ζ, µ > 0 and K > 0, we say that property Πζ, µ, K holds for R = R 1,..., R 1 Ω f and Ṽ = ṼK = {v V : d R v Kpm, 1 1} BR = {b = v 1,..., v B : v Ṽ v j, v R j, 1 j 1} satsfy the nequaltes Ṽ 1 µm, 19 BR ζp 1 m. 0 We thn of BR as the members of B that have been pced-up by the random element R Ω. We wll be nterested n the probablty that the property Πζ, µ, K fals for a fxed B n the unform probablty space Ω.

14 14 Y. Kohayaawa, V. Rödl, and M. Schacht Lemma 4.3 Pc-Up Lemma. For every β, ζ and µ wth 1 > β, ζ, µ > 0 there exst 1 > η = ηβ, ζ, µ > 0, K = Kβ, µ > 0 and m 0 such that f m m 0 and B ηm, 1 then PΠζ, µ, K fals for R Ω β 1T. For the proof we need a few defntons. Suppose β and µ are gven. We defne θ = 1 β 1, 3 { } 3 1 log 1/θ K = max, e. 4 µ Snce p 1/m the defnton of K 3 1 log1/θ/µ mples that m µt K log K 1 exp θ T 5 µm/ 1 1 holds for m suffcently large. Usng the defnton of d R n 18 we construct for each = 1,..., 1 a subset of V by puttng V = {v V 1 : d R v Kpm}, where V 0 = V. Observe that V = V 0 V 1 V 1 Lemma 4.3 we defne the followng bad events n Ω. = Ṽ. In the vew of Defnton 4.4 A, B. For each = 0,..., 1 and K, µ > 0, ζ > 0, let A = A µ, K, B = Bζ, K Ω be the events Observe that the defnton of V 0 A : V < 1 µ/ 1 m, B : BR > ζp 1 m. = V mples PA 0 = 0. 6 We restate Lemma 4.3 by usng the notaton ntroduced n Defnton 4.4. Lemma 4.3 Pc-up Lemma, event verson. For every β, ζ and µ wth 1 > β, ζ, µ > 0 there exst 1 > η = ηβ, ζ, µ > 0, K = Kβ, µ > 0 and m 0 such that f m m 0 and then B ηm, 7 PA 1 µ, K Bζ, K β 1T. 8

15 The Turán Theorem for Random Graphs 15 We need some more preparaton before we prove Lemma 4.3. Suppose β, ζ, µ are gven by Lemma 4.3 and θ, K are fxed by 3 and 4. For each = 1,..., 1 we consder the set B B consstng of those -tuples b B whch were partally pced up by edges of R 1,..., R. For techncal reasons we consder only those -tuples contanng vertces v V 1,.e., wth d Rj v Kpm for j = 1,..., 1. More formally, we let B = {b = v 1,..., v B : v V 1 We also set B 0 = B. The defntons of Ṽ = V 1 V and B 1 mply v j, v R j, 1 j }. BR B 1. 9 Equalty may fal n 9 because we may have V \V 1. For each =,..., 1 defne ζ 1 by ζ 1 = ζ, ζ 1 = 1 1µ 4 1K 1 ζ θ 4K /ζ. 30 Furthermore, consder for each = 0,..., 1 the event B = B ζ, K Ω defned by B : B > ζ p m. 31 In order to prove Lemma 4.3 we need two more clams, whch we wll prove later. Clam 4.5. Clam 4.6. For all 1 1, we have PA = P V < 1 µ m θ T. 1 For all 1 1, we have PB A 1 B 1 θ T. Assumng Clams 4.5 and 4.6, we may easly prove Lemma 4.3. Proof of Lemma 4.3. Set η = ζ 0 where ζ 0 s gven by 30. The defnton of B 0 = B and 7 mples B 0 ζ 0 m and consequently by the defnton of the event B 0 n 31 PB 0 = 0. 3 Because of 9 and ζ 1 = ζ n 30 we have PB PB Usng the formal dentty PB = PB A 1 B 1 + PB A 1 B 1, we observe that PB PB A 1 B 1 + PA 1 + PB 1 34

16 16 Y. Kohayaawa, V. Rödl, and M. Schacht for each = 1,..., 1. It follows by applyng 33 and 34 that PA 1 B PA 1 + PB 1 1 PA 1 + PB A 1 B 1 + PA 1 + PB 0. =1 Clams 4.5 and 4.6, and 6, 3 and 3 fnally mply β PA 1 B 1θ T 1 T 1 β 1T for m suffcently large, as requred. We now prove Clam 4.5 and then Clam 4.6. Proof of Clam 4.5. Fx a set V V of sze µm/ 1. For a fxed j 1 j assume that d Rj v > Kpm for every v n V. Ths clearly mples the event E j V : R j V j V > Kpm µm 1 = K µt The T pars of R j are chosen unformly n V j V, so the hypergeometrc tal lemma, Lemma 4.1, apples, and usng the fact that e K 1/ by 4 we get e KµT / 1 P E j V µt K log K exp. 36 K 1 Set E j = E j V, where the unon s taen over all V V of sze µm/ 1. Then m µt K log K PE j exp 37 µm/ 1 1 holds for each j = 1,...,, and ths mples m P E j µm/ 1 j=1 exp µt K log K. 1 Fnally, the fact that A j=1 E j and the choce of K wth 5 gves that m µt K log K PA exp θ T, µm/ 1 1 as requred. Proof of Clam 4.6. Recall β, ζ and µ are gven by Lemma 4.3 and θ, K and ζ are fxed by 3, 4 and 30. In order to prove Clam 4.6 we fx 1 1 and we assume A 1 and B 1 occur. Ths means by Defnton 4.4 and 31 that V 1 1µ 1 1µ 1 m = m, B 1 ζ 1 p 1 m. 39

17 The Turán Theorem for Random Graphs 17 We have to show that B ζ p m 40 holds for R n the unform probablty space Ω wth probablty 1 θ T. Frst we defne the auxlary constant L = 4K 1 1 /ζ. 41 θ The defnton of θ n 3 and the facts that 0 < ζ < 1 for each = 1,..., 1 and K > 1 mply that 4 L β 1 > e 4 holds. We defne the degree of a par n V V 1 wth respect to B 1 by d B 1 w, w = {b = v 1,..., v B 1 : v = w and v = w }. We can bound the value of the average degree by 38 and 39: { } avg d B 1 v, v : v, v V V 1 = B 1 m V µ ζ 1p 1 m. 43 We also can bound B 1 V, V 1 = max{d B 1 v, v : v, v V V 1 } by the followng observaton. Let v, v be an arbtrary element n V V 1. Then, by the defnton of V 1, we have d B 1 v, v d R1 v... d R 1 v m 1 Kpm 1 m Inequalty 44 mples B 1 V, V 1 K 1 p 1 m. 45 Let F be the set of pars of hgh degree. More precsely, set { F = v, v V V 1 : d B 1 > ζ } p 1 m. A smple averagng argument applyng 43 yelds 1ζ 1 F V V 1 1ζ 1 m µζ 1 1µζ

18 18 Y. Kohayaawa, V. Rödl, and M. Schacht On the other hand, f we set F = V V 1 \ F then the defnton of F and 45 mply B = d B 1 v, v + d B 1 v, v Next we prove that = v,v R F ζ v,v R F p 1 m R F + K 1 p 1 m R F ζ p 1 m T + K 1 p 1 m R F ζ + K 1 T R F p m. 47 whch, together wth 47, yelds our clam, namely, that P R F > ζ T K 1 θ T, 48 P B > ζ p m θ T. 49 We now prove nequalty 48. Wthout loss of generalty we assume equalty holds n 46. Then the hypergeometrc tal lemma, Lemma 4.1, mples that F T 1ζ 1 P R F > L m = P R F > L T 1 1µζ 1ζ 1 L T e 1 1µζ 50 L exp L log L 1ζ 1 T, 1 1µζ where n the last nequalty we used that L e see 4. The defntons of ζ 1 and L n 30 and 41 yeld We use the last nequalty to derve L 1ζ 1 = L ζ 1 /ζ θ4k = ζ 1 1µζ 4K 1 4K 1. L log L 1ζ 1 1 1µζ = log 1 θ, L 1ζ 1 = 1 1µζ whch, combned wth nequalty 50, gves 48. ζ K 1, 4.. The -tuple lemma for subgraphs of random graphs Let G Gn, q be the bnomal random graph wth edge probablty q = qn, and suppose H = U, W ; F s a bpartte, not necessarly nduced subgraph of G wth U = m 1 and W = m. Furthermore, denote the densty of H by p = eh/m 1 m. We now consder subsets of W of fxed cardnalty 1, and classfy them accordng

19 The Turán Theorem for Random Graphs 19 to the sze of ther jont neghbourhood n H. For ths purpose we defne B U, W ; γ = { b = {v 1,..., v } W : d H U b p m 1 γp } m 1, where d H U b denotes the sze of the jont neghbourhood of b n H, that s, d H U b = Γ H v. =1 The followng lemma states that n a typcal G Gn, q the set B U, W ; γ s small for any suffcently large ε, q-regular subgraph H = U, W ; F of a dense enough random graph G. Recall that f G s a graph and U, W V G are two dsjont sets of vertces, then G[U, W ] denotes the bpartte graph naturally nduced by U, W. Lemma 4.7 The -tuple lemma. For any constants α > 0, γ > 0, η > 0, and 1 and functon m 0 = m 0 n such that q m 0 log n 4, there exsts a constant ε > 0 for whch the random graph G Gn, q satsfes the followng property wth probablty 1 o1: If for a bpartte subgraph H = U, W ; F of G the condtons eh αeg[u, W ], H s ε, q-regular, U = m 1 m 0 and W = m m 0 apply, then also apples. B U, W ; γ η Proof. The proof of Lemma 4.7 s gven n [11]. m 4.3. Outlne of the proof of the countng lemma for l = 4 The proof of the Lemma.3 contans some techncal defntons. In order to mae the readng more comprehensble, we frst nformally llustrate the basc deas of the proof for the case l = 4, before we gve the proof for a general l n Secton 4.4. Consder the followng stuaton: Let V 1, V, V 3, and V 4 be parwse dsjont sets of vertces of sze m. Let J be a 4-partte graph wth vertex set V J = V 1 V V 3 V 4. We thn of J as a not necessarly nduced subgraph of a random graph n Gn, q wth T = pm edges between each V and V j 1 < j 4, where p = αq. We wll descrbe a stuaton n whch we wll be able to assert that J contans the rght number of K 4 s. Here and everywhere below by the rght number we mean as expected n a random graph of densty p ; notce that, for the number of K 4 s, ths means p 6 m 4. Observe that, however, J s a not necessarly nduced subgraph of a graph n Gn, q, and ths maes our tas hard. As t turns out, t wll be more convenent to magne that J s generated n l 1 = 3 stages. Frst we choose the edges from V 4 to V 1 V V 3. Then we choose the edges from V 3 to V 1 V, and n the thrd stage we dsclose the edges between V and V 1. The ey dea of the proof s to consder bad tuples, whch we create n every stage. 51

20 V v 0 Y. Kohayaawa, V. Rödl, and M. Schacht bad 3-tuples V 3 Ṽ 3 v 3 dscarded vertces V 1 v 1 pced-up 3-tuple Par v 1, v s good f t has: approxametly expected number of jont neghbours v 3 such that v 1, v, v 3 s not a bad 3-tuple a b Fgure 1 After we chose the edges from V 4 to the other vertex classes, we defne bad 3-tuples n V 1 V V 3 : a 3-tuple s bad f ts jont neghbourhood n V 4 s much smaller than expected. Then, wth the rght choce of constants, Proposton 4.11 for = 3 and J = J[V 4, V 1 V V 3 ] wll ensure that there are not too many bad 3-tuples. Proposton 4.11 s a corollary of the the -tuple lemma, Lemma 4.7. We next generate the edges between V 3 and V 1 V. We want to defne bad pars n V 1 V. Here t becomes slghtly more complcated to dstngush bad from good. Ths s because there are two thngs that mght go wrong for a par n V 1 V. Frst of all, agan the jont neghbourhood now n V 3 of a par n V 1 V mght be too small. On the other hand, t could have the rght number of jont neghbours n V 3, but many of these neghbours complete the par to a bad 3-tuple. Here the Pc-Up Lemma comes nto play for = 3 see Proposton 4.10: ths lemma wll ensure that, gven the set of bad 3-tuples whch was already defned n the frst stage s small, we wll not pc-up too many of these see Fgure 1a, whle choosng the edges between V 3 and V 1 V. We say that a trple v 1, v, v 3 has been pced-up f v 1, v 3 and v, v 3 are n the edge set generated between V 3 and V 1 V. Here the stuaton complcates somewhat. The Pc-Up Lemma forces us to dscard a small porton less or equal µ PU fracton of vertces n V 3. Thus, n order to avod the frst type of badness too small jont neghbourhood as a -tuple n V 1 V t s not enough to have the rght number of jont neghbours n V 3 ; we need the rght number of jont neghbours n Ṽ3, whch s V 3 wthout the µ PU m vertces at most we lose by applyng the Pc-Up Lemma see Fgure 1b. Ths wll be ensured by the the -tuple lemma to be more precse, Proposton 4.11, now for = and J = J[Ṽ3, V 1 V ]. Later, n the general case, we wll refer to the set of bad -tuples n V 1 V

21 The Turán Theorem for Random Graphs 1 as B see Defnton 4.8 below. We defne B as the unon of the sets B a and B b, defned as follows. We put n B a the -tuples that are bad because they have a jont neghbourhood n Ṽ+1 that s too small; the set B b s defned as the set of -tuples n V 1 V that bad because they extend to too many bad + 1-tuples.e., + 1-tuples n B +1. As descrbed above, we defne B = l 1,..., 1 by reverse nducton, startng wth B l 1, and gong down to B 1. Wth the rght choce of constants, there wll not be too many bad vertces n V 1. Havng ensured that most of the m vertces n V 1 are not bad.e., do not belong to B 1 we are now able to count the number of K 4 s. We wll use the followng determnstc argument, whch wll later be formalzed n Lemma Consder a vertex v 1 n V 1 that s not bad. Ths vertex has approxmately the expected number of neghbours n Ṽ.e., pm, and not too many of these neghbours consttute, together wth v 1, a bad - tuple. In other words, ths means that v 1 extends to pm copes of K n V 1 V \ B. Ths mples that each such K has the rght number of jont neghbours n Ṽ3.e., p m, and consequently extends to the rght number of K 3 s n V 1 V V 3 \ B 3. Repeatng the last argument, each of these K 3 s extends nto p 3 m dfferent copes of K 4. Snce we have ensured that most of the m vertces n V 1 are not bad, we have m pm p m p 3 m = p 4 m 4 copes of K Proof of the countng lemma In ths secton we wll prove Lemma.3. In the secton Concepts and Constants, we ntroduce the ey defntons and descrbe the logc of all mportant constants whch wll appear later n the proof. Afterwards we prove two techncal propostons n the secton Tools. These propostons correspond to the lemmas n Sectons 4.1 and 4., and ther use wll gve a short proof of the Countng Lemma, to be presented n the secton Man proof. Concepts and constants. Let t l be fxed ntegers and let n be suffcently large. Let α and ε be postve constants. Let G Gn, q be the bnomal random graph wth edge probablty q = qn, and suppose J s an l-partte subgraph of G wth vertex classes V 1,..., V l. For all 1 < j l we denote by J j the bpartte graph nduced by V and V j. Consder the followng assertons for J. I V = m = n/t for all 1 l, II q l 1 n log n 4, III J j 1 < j l has T = pm edges, where 1 > αq = p 1/n, and IV J j 1 < j l s ε, q-regular. Let σ > 0 be gven. We defne the constants γ = µ = ν = σ 1/l, 5 3 and, for 1 l, we put β +1 = 1 α l 1/. 53 e

22 Y. Kohayaawa, V. Rödl, and M. Schacht In order to prove Lemma.3 we need some defntons. These defntons always depend on a fxed subgraph J of our random graph G Gn, q satsfyng I IV. However, we wll drop references to J because we want to smplfy the notaton e.g., we wrte V nstead of V J. Also, for each = 1,..., l we denote V 1 V by W. In the proof we consder for a fxed J sets of bad -tuples B W 1 l 1. We defne these sets recursvely from B l 1 to B 1. As mentoned above n the dscusson of the l = 4 case, there are two reasons that mae a gven -tuple n W bad. Frst of all, ts jont neghbourhood n V +1 mght be too small see the defnton of B a n Defnton 4.8 and, secondly, t could extend nto too many bad + 1-tuples n B +1 see the defnton of B b n Defnton 4.8. Note that the bad + 1-tuples have already been defned, as we are usng reverse nducton n these defntons. Next we apply the Pc-Up Lemma for = l wth µ PU +1 = µ and β+1 PU = β +1 and yet unspecfed ζ+1 PU. As a result we obtan KPU +1 = KPU +1 βpu +1, µpu +1 and the set of undscarded vertces wth Ṽ +1 = Ṽ PU +1K PU +1 V +1 Ṽ+1 1 µm. We need a few more defntons before we defne B, B a and B b recursvely for = l 1,..., 1. Let Γ +1 b be the jont neghbourhood of b = v 1,..., v W n Ṽ+1 wth respect to J, more precsely Γ +1 b = {w Ṽ+1 : v j, w EJ j,+1, 1 j }. For a fxed set B W +1 and b = v 1,..., v W we denote the degree d B b of b n B wth respect to J by { d B b = w Γ +1 b: v 1,..., v, w B}. Next we defne stll for a fxed J the sets of bad -tuples B = B γ, µ, ν W mentoned earler. Although we do not apply the Pc-Up Lemma for = l, for the sae of convenence we consder the neghbourhood of elements n W l 1 n Ṽl, nstead of n V l., B b, B. Let γ, µ, ν be gven by 5. We defne recur- Defnton 4.8 B l 1, B a svely the followng sets of bad tuples for = l 1,..., 1: B l 1 = B l 1 γ, µ = B a = B a γ, µ = B b = B b { } b W l 1 : Γ l b < 1 γ µp l 1 m, { } b W : Γ +1 b < 1 γ µp m, ν = { b W : d B+1 b νp m }, γ, µ B b ν. B = B γ, µ, ν = B a We also consder bad events n Gn, q defned on the bass of the sze of the sets B l 1 γ, µ, B a γ, µ, B b ν, and B γ, µ, ν defned above. In the followng defnton we mean by J an arbtrary subgraph of G Gn, q satsfyng condtons I IV.

23 The Turán Theorem for Random Graphs 3 Defnton 4.9. Let γ, µ, ν be gven by 5 and let η > 0 = l 1,..., 1 be fxed. We defne the events X l 1 γ, µ, η l 1 : J G s.t. B l 1 > η l 1 /m l 1, X a γ, µ, η : J G s.t. B a > η /m, X b γ, µ, ν, η, η +1 : J G s.t. B +1 η +1 m +1 B b > η /m, X γ, µ, ν, η, η +1 = X a γ, µ, η X b γ, µ, ν, η, η +1. For smplcty, we let and X b X a X a l 1 = X l 1 = X l 1 γ, µ, η l 1, = X a γ, µ, η for = 1,..., l 1, = X b γ, µ, ν, η, η +1 for = 1,..., l, X = X γ, µ, ν, η, η +1 for = 1,..., l 1. Owng to the specal role of X 1 later n the proof, we let X bad = X bad γ, µ, ν, η 1, η = X 1 γ, µ, ν, η 1, η. We wll now descrbe the remanng constants used n the proof. Notce that α and σ were gven and we have already fxed γ, µ, and ν n 5 and β for l 1 n 53. The yet unspecfed parameters η and ε wll be determned by Propostons 4.10 and Frst we set η 1 = ν. Then Proposton 4.10 PU +1 nductvely descrbes η +1 = η +1 β +1, γ, µ, ν, η for = 1,..., l such that PX b = o1. Fnally, for = 1,..., l 1, Proposton 4.11 TL mples the choce for ε = ε α, γ, µ, η such that = o1. We set PX a ε = mn{ε : = 1,..., l 1}. A dagram llustratng the defnton scheme for the constants above s gven n Fgure. α, σ, γ, µ, ν, β,..., β l 1 η 1 = ν PU η η PU +1 η +1 PU l 1 η l 1 TL 1 TL TL l 1 } ε 1 ε... ε {{ ε ε l 1 } ε = mn ε Fgure Flowchart of the constants Thus, ε s defned for any gven α and σ, as clamed n Lemma.3. From now on, these constants are fxed for the rest of the proof of Lemma.3.

24 4 Y. Kohayaawa, V. Rödl, and M. Schacht Tools. We need some auxlary results before we prove Lemma.3. For ths purpose we state varants of the Pc-Up Lemma, Lemma 4.3, and of the -tuple lemma, Lemma 4.7, n the form that we apply these later. These varants wll be referred to as PU +1 and TL. The next proposton follows from Lemma 4.3 for = l. Proposton 4.10 PU +1. Fx 1 l. Let α, σ > 0 be arbtrary, let γ, µ, ν and β +1 be gven by 5 and 53, and let η be defned as stated n Secton 4.4 see Fgure. Then there exsts η +1 = η +1 β +1, γ, µ, ν, η > 0 such that for every t l a random graph G n Gn, q satsfes the followng property wth probablty 1 o1: If J s a subgraph of G satsfyng I IV and B +1 γ, µ, ν W +1 s such that then the number of -tuples b n W wth B +1 γ, µ, ν η +1 m +1, 54 d B+1 b νp m s less than whch means Furthermore, holds. B b η m, ν η m. 55 Ṽ+1 1 µm We restate Proposton 4.10, by usng the events X b that nequaltes 54 and 55 correspond to X b to the frst part of Proposton from Defnton 4.9. Observe = o1 s equvalent, so that PX b Proposton 4.10 PU +1. Fx 1 l. Let α, σ > 0 be arbtrary, let γ, µ, ν and β +1 be gven by 5 and 53, and let η be defned as stated n Secton 4.4 see Fgure. Then there exsts η +1 = η +1 β +1, γ, µ, ν, η > 0 such that for every t l P X b γ, µ, ν, η, η +1 = o1 and P Ṽ+1 < 1 µm = o1. Proof. We apply Lemma 4.3 for = + 1 and wth the followng choce of β PU, ζ PU,

25 The Turán Theorem for Random Graphs 5 µ PU : β PU = β +1, 56 ζ PU = η ν, 57 µ PU = µ. 58 Lemma 4.3 then gves η PU, from whch we defne the constant η +1 we are loong for by puttng η +1 = η PU. We assume nequalty 54 holds. In other words, the number of the bad + 1-tuples n W +1 s B +1 η +1 m +1 = η PU m On the other hand, f we assume that 55 does not hold.e., the event X b occurs, then the number of + 1-tuples n B +1 that have been pced-up has to exceed η m νp m = ζ PU p m The Pc-Up Lemma bounds the number of these confguratons n V1 V +1 V V +1 T T by β PU T m = β +1 T m. 61 T T We now estmate the number of all possble graphs J satsfyng I IV for whch 59 holds but the number of members n B +1 that have been pced-up exceeds 60. There are fewer than n l m dfferent ways to fx the l vertex classes of J. Furthermore, observe that B +1 s determned by all the edges n J jj < j l, 1 j < j l, whch +1 dfferent pars jj. Thus we have at most m l +1 possbltes to gves l T determne B +1. Ths, combned wth 61, III, and 53, yelds that P X b n m l m T nl em q T l +1 β +1 T m q l T T l T β +1 T e nl α Snce l s fxed and T m = n/t, we have P = o1. X b l β+1 T nl T. Note that the set Ṽ+1 was determned by the applcaton of the Pc-Up Lemma. Therefore, the second asserton n Proposton 4.10 also follows from the proof above.

26 6 Y. Kohayaawa, V. Rödl, and M. Schacht The followng s an easy consequence of Lemma 4.7 for = 1 l 1. Proposton 4.11 TL. Fx 1 l 1. Let α, σ > 0 be arbtrary, let γ, µ be gven by 5, and let η be defned as stated n Secton 4.4 see Fgure. Then there exsts ε = ε α, γ, µ, η > 0 such that for every t l a random graph G n Gn, q satsfes the followng property wth probablty 1 o1: If ε ε and J s a subgraph of G satsfyng I IV, then the number of -tuples b n W wth Γ +1 b < 1 γ µp m s less than whch means that η m, B a γ, µ η m. 6 We can reformulate Proposton 4.11 n a shorter way by usng the event X a Defnton 4.9. see Proposton 4.11 TL. Fx 1 l 1. Let α, σ > 0 be arbtrary, let γ, µ be gven by 5 and let η be defned as stated n Secton 4.4 see Fgure. Then there exsts ε = ε α, γ, µ, η > 0 such that for every t l and ε ε P X a γ, µ, η = o1. Proof. We apply the -tuple lemma, Lemma 4.7, wth =, α TL = α/3, γ TL = γ and η TL = η /. 63 The -tuple lemma gves an ε TL and we set ε = mn{ ε TL 3, α/, 1/7}. Let ε ε and J be a subgraph of G Gn, q satsfyng I IV. Set U = Ṽ+1 and W = j=1 V j. By IV, the graph J jj 1 j < j s ε, q-regular. A straghtforward argument usng ε 1/7 and Lemma 3. for C = 3/ shows that wth probablty 1 o1 the subgraph J[U, W ] s at least 3 ε, q-regular and therefore ε TL, q-regular, whch yelds condton of Lemma 4.7. Moreover, wth probablty 1 o1 we have, say, and usng the regularty of J we see that EG[U, W ] 3 q1 µm, EJ[U, W ] α εq1 µm, whch by our choce of ε gves condton of Lemma 4.7. Fnally, wth asserton II for J all assumptons of the -tuple lemma are satsfed for J[U, W ]. Therefore, the -tuple lemma mples that, wth probablty 1 o1, we have { b W : } Γ +1 b 1 γp 1 µm η TL m.

27 The Turán Theorem for Random Graphs 7 The choce of η TL n 63 gves { } b W : Γ +1 b 1 γ µ + γµp η m m, and hence 6 holds wth probablty 1 o1, by the smple observaton that Γ +1 b 1 γ µp m mples Γ +1 b 1 γ µ + γµp m. Man proof. Our proof of the Countng Lemma, Lemma.3, follows mmedately from Lemmas 4.1 and 4.13 below. Lemma 4.1 s a probablstc statement and asserts that the probablty of the event X bad Gn, q s o1. On the other hand, Lemma 4.13 s determnstc and clams that f a graph G s not n X bad and J s a not necessarly nduced subgraph of G satsfyng I IV, then J contans the rght number of copes of K l. We apply the techncal propostons from the last secton n the proof of the probablstc Lemma 4.1 below. Lemma 4.1. For arbtrary α and σ > 0, let γ, µ, ν be gven by 5, and let ε and η =,..., l 1 be defned as stated n Secton 4.4. Let G be a random graph n Gn, q. Then Proof. Formal logc mples PG X bad γ, µ, ν = o1. X bad X a 1 X b 1 X X a X b X 3.. X a l X b l X l 1 X l 1, and thus, by Propostons 4.10 and 4.11 notce X l 1 = X a l 1 by Defnton 4.9, we have l P X bad =1 PX a + PX b + PX l 1 = o1. Lemma For arbtrary α and σ > 0, let γ, µ, ν be gven by 5, and let ε and η =,..., l 1 be defned as stated n Secton 4.4. Then every subgraph J of a graph G X bad γ, µ, ν satsfyng condtons I IV contans at least copes of K l. 1 σp l m l Proof. We shall prove by nducton on that the followng statement holds for all 1 l:

28 8 Y. Kohayaawa, V. Rödl, and M. Schacht S Let J be a subgraph of G X bad such that I IV apply. Then there are at least 1 γ µ ν p m dfferent -tuples n W \ B that nduce a K n J[V 1,..., V ]. Suppose = 1. Note that X bad mples that V 1 B 1 η 1 m = νm. Therefore V 1 \ B 1 contans at least 1 νm 1 γ µ νp 0 m 1 copes of K 1. We now proceed to the nducton step. Assume and S 1 holds. Therefore, W 1 \ B 1 contans at least 1 γ µ ν 1 p 1 m 1 dfferent 1-tuples b = v 1,..., v 1, each consttutng the vertex set of a K 1 n J[V 1,..., V 1 ]. For every b W 1 \ B 1, we have Γ b 1 γ µp 1 m, and d B b < νp 1 m. Therefore, every such b extends to at least 1 γ µ νp 1 m dfferent b W \ B that correspond to a K J[V 1,..., V ]. Ths mples S, and hence our nducton s complete. Asserton S l and the choce of γ, µ, and ν n 5 gve at least copes of K l n J. 1 γ µ ν l p l m l = 1 σp l m l Clearly, Lemmas 4.1 and 4.13 together mply the Countng Lemma, Lemma The d-degenerate case In ths secton we descrbe how the proof of Theorem 1. extends to the proof of Theorem 1.. The detaled proof of Theorem 1. s gven n [14]. Frst we outlne the proof of Theorem 1., assumng a counterpart for the Countng Lemma, Lemma.3, whch we state below. Let d be an nteger and H a d-degenerate graph on h vertces. Let t h be fxed ntegers and let n be suffcently large. Let α and ε be constants greater than 0. Suppose J s an h-partte subgraph of G wth vertex classes V 1,..., V h satsfyng the followng condtons: I V = m = n/t for all, II q d n log n 4, III for all 1 < j h, EJ j = where 1 > αq = p 1/n, and IV J j 1 < j h s ε, q-regular. { T = pm f {w, w j } EH 0 f {w, w j } EH, We now state the approprate countng lemma for the d-degenerate case. Lemma.3 Countng lemma, d-degenerate case. For every α, σ > 0, nteger d and d-degenerate graph H on h vertces, there exsts ε > 0 such that for every t h a

29 The Turán Theorem for Random Graphs 9 random graph G n Gn, q satsfes the followng property wth probablty 1 o1: Every subgraph J G satsfyng condtons I IV contans at least copes of H. 1 σp EH m h Setch of the proof of Theorem 1.. Let d be a fxed postve nteger and suppose H s a d-degenerate graph of order h. Let the vertces of H be ordered w 1,..., w h such that each w has at most d neghbours n {w 1,..., w 1 }. At frst, we follow the proof of Theorem 1. and observe that, by 16, the Erdős Stone Smonovts theorem see 1 mples that F cluster contans at least one copy of H f we choose t SRL 0 bg enough. Ths yelds, n the same way as n the orgnal proof, that F contans an h-partte ε Lem.3, q-regular graph J wth EJ j = α Lem.3 qm f {w, w j } EH and EJ j = f {w, w j } EH. For 1 h, we dentfy the vertex class V n J wth the vertex w V H. We then apply Lemma.3 wth approprate α Lem.3 and 0 < σ < 1 to deduce Theorem 1.. Fnally, we outlne of the proof of Lemma.3. Setch of the proof of Lemma.3. We prove Lemma.3 n the same way as Lemma.3. Observe that condtons I and IV are unchanged n Lemma.3. Condtons III and III state that J s a blown-up copy of the subgraph we are consderng, namely, K l and H, respectvely. The man dfference s between II and II. The crucal part of the proof of the orgnal countng lemma s the defnton of bad tuples n Defnton 4.8. Recall that the proof of Lemma.3 used the Pc-Up Lemma Lemma 4.3. There we had to dscard a small porton of the vertces of V of hgh degree to some V j, j < to obtan Ṽ V. For 1 V K l, we consdered two types of bad 1-tuples n W 1 = V 1 V 1. The frst type, the ones put n B a 1, was determned by the sze of ther jont neghbourhood n Ṽ. On the other hand, an 1-tuple n W 1 was bad of the second type, and was put n B b 1, f t was contaned n too many bad -tuples n B. We use the property that H s d-degenerate to change the defnton of B a. In the proof of Lemma.3 we wanted to extend nductvely each K 1 n W 1 that s not bad to the rght number of copes of K n W \B. For ths purpose we had to consder the jont neghbourhood of all vertces n the 1-tuple. The graph H s d-degenerate, and we fxed an orderng w 1,..., w h of V H so that each w has at most d neghbours n {w 1,..., w 1 }. Ths mples that t s suffcent to consder the jont neghbourhood of at most d elements of the 1-tuple to determne ts badness, or ts membershp n B a 1. For = 1,..., h, we defne the ndex sets I consstng of the the ndces of the neghbours of w n {w 1,..., w 1 }. Also, for a fxed 1-tuple v 1,..., v 1 W 1, we consder the jont neghbourhood of Γv j Ṽ =: Γvj, where the ntersecton

30 30 Y. Kohayaawa, V. Rödl, and M. Schacht s taen over j I. More precsely, we defne B a as follows: I = {j [ 1]: {w j, w } EH}, B a 1 γ, µ = v 1,..., v 1 W 1 : Γ v j < 1 γ µp I m. j I Obvously, I d for 1 h 64 holds. The defnton of B b remans almost unchanged; agan, for some B W +1 and b = v 1,..., v W, we set d B b = {w Γ +1 : v 1,..., v, w B} and we only adjust the exponent of p: B b = B b ν = { } b W : d B+1 b νp I m. Then we defne the correspondng events exactly as n Defnton 4.9. The proof of Lemma.3 conssts of two propostons Propostons 4.10 and 4.11 and two lemmas Lemmas 4.1 and We now dscuss the proofs of the correspondng results wth the new defnton for the famles B a and B b under I IV nstead of I IV, and wth K l replaced by an arbtrary d-degenerate graph H. We defne the followng constants, slghtly dfferent compared to the ones n the orgnal proof see 5 and 53: γ = µ = ν = σ 1/h, 65 3 and, for 1 l and I +1 > 0, β +1 = 1 α h Ij 1/ I +1 j=. 66 e The other constants are defned n the same way as descrbed n Secton 4.4 see Fgure, wth l replaced by h. We now dscuss the proofs of the results that correspond to Propostons 4.10 and 4.11 and Lemmas 4.1 and Proposton The proof s an applcaton of the Pc-Up Lemma, Lemma 4.3, for = + 1. The Pc-Up Lemma does not requre condton II. It s already vald for qn 1/n, whch s stll guaranteed by II. It s easy to see that X b s mpossble f we set η +1 = η ν/ and f I +1 = 0. If I +1 > 0, then essentally the same calculaton wth the new β +1 defned n 66 gves the proposton. We apply the Pc-Up Lemma for the space Vj V +1 j I +1 T and the projecton of B+1 onto j I V +1 j V +1. Proposton The proof s a straghtforward applcaton of the -tuple lemma, Lemma 4.7. In the orgnal proof we apply the -tuple lemma for = 1 l 1 and we needed condton II namely, q l 1 n log n 4 for = l 1. Here, the new defnton of B a 1 from above comes nto play. Inequalty 64 ensures that we consder at most the jont neghbourhood of d vertces. Ths means that we apply the -tuple lemma for d and thus condton II namely, q d n log n 4 s suffcent.