Ramsey-type theorems with forbidden subgraphs

Transcription

1 Ramsey-type theorems with forbidde subgraphs Noga Alo Jáos Pach József Solymosi Abstract A graph is called H-free if it cotais o iduced copy of H. We discuss the followig questio raised by Erdős ad Hajal. Is it true that for every graph H, there exists a ε(h) > 0 such that ay H-free graph with vertices cotais either a complete or a empty subgraph of size at least ε(h)? We aswer this questio i the affirmative for a special class of graphs, ad give a equivalet reformulatio for touramets. I order to prove the equivalece, we establish several Ramsey type results for touramets. 1 Itroductio Give a graph G with vertex set V (G) ad edge set E(G), let α(g) ad ω(g) deote the size of the largest idepedet set (empty subgraph) ad the size of the largest clique (complete subgraph) i G, respectively. A subset U V (G) is called homogeeous, if it is either a idepedet set or a clique. Deote by hom(g) the size of the largest homogeeous set i G, i.e., let hom(g) = max (α(g), ω(g)). If H is ot a iduced subgraph of G, the we say that G is a H-free graph. Accordig to classical Ramsey theory, hom(g) 1 2 log 2 for every graph G with vertices [ES35], ad there exists some G with hom(g) < 2 log 2 (see [E47]). Erdős ad Hajal [EH89] raised the possibility that the followig could be true. Departmet of Mathematics, Raymod ad Beverly Sackler Faculty of Exact Scieces, Tel-Aviv Uiversity, Tel- Aviv, Israel. oga@math.tau.ac.il. Supported by a USA Israeli BSF grat, by a grat from the Israel Sciece Foudatio ad by the Herma Mikowski Mierva Ceter for Geometry at Tel Aviv Uiversity. Mathematical Istitute of the Hugaria Academy of Scieces H-1364 Budapest, P.O.B Supported by NSF grat CR , PSC-CUNY Research Award , ad OTKA-T Computer ad Automatio Istitute of the Hugaria Academy of Scieces H-1518 Budapest, P.O.B. 63. Supported by TKI grat Stochastics@TUB, ad OTKA-T

2 Cojecture 1 For every graph H, there exists a positive ε = ε(h) such that every H-free graph with vertices has a homogeeous set whose size is at least ε. Erdős ad Hajal cofirmed their cojecture for every graph H which belogs to the class H defied recursively as follows: 1. K 1, the graph cosistig of a sigle vertex, belogs to H; 2. if H 1 ad H 2 are two vertex-disjoit graphs belogig to H, the their disjoit uio as well as the graph obtaied from this uio by coectig every vertex of H 1 to every vertex of H 2 belogs to H. Gyárfás [G97] oticed that it follows from a well kow result of Seische [Se74] that Cojecture 1 is also true for all graphs geerated by the above rules startig with P 4, a simple path with 4 vertices, ad K 1. Our first theorem exteds both of these results. If Cojecture 1 is true for some graph H, the we say that H has the Erdős-Hajal property. For ay graph H with vertex set V (H) = {v 1,..., v k } ad for ay other graphs, F 1,..., F k, let H(F 1,..., F k ) deote the graph obtaied from H by replacig each v i with a copy of F i, ad joiig a vertex of the copy of F i to a vertex of a copy of F j, j i, if ad oly if v i v j E(H). The copies of F i, i = 1,..., k, are supposed to be vertex disjoit. Theorem 1.1 If H, F 1,..., F k have the Erdős-Hajal property, the so does H(F 1,..., F k ). I other words, the Erdős-Hajal property is preserved by replacemet. This eables us to verify that Cojecture 1 is true, e.g., for the graphs depicted i Figure 1, which aswers some questios of Gyárfás [G97]. Figure 1: G 1 ad G 2 have the Erdős-Hajal property No o-perfect graph is kow to have the Erdős-Hajal property. Ufortuately, i this respect Theorem 1.1 caot offer ay help. Ideed, accordig to a result of Lovász [L83], which played a key role i his proof of the Weak Perfect Graph Cojecture [L72], perfectess is also preserved by replacemet. It is a outstadig ope problem to decide whether the smallest o-perfect graph, the cycle of legth 5, has the Erdős-Hajal property. As Lovász poited out, there is a eve simpler usolved Problem Does there exist a positive costat ε so that, for every graph G o vertices such that either G or its complemet Ḡ cotais a iduced odd cycle whose legth is at least 5, we have hom(g) ε? 2

3 It is easy to formulate aalogous questios for touramets. A touramet with o directed cycle is called trasitive. If a touramet has o subtouramet isomorphic to T, the it is called T -free. It is well kow [EEH73],[S74] that every touramet of vertices cotais a trasitive subtouramet whose size is at least c log, ad that this result is tight apart from the value of the costat. Cojecture 2 For every touramet T, there exists a positive ε = ε(t ) such that every T -free touramet with vertices has a trasitive subtouramet whose size is at least ε. Theorem 1.2 Cojecture 1 ad Cojecture 2 are equivalet. I order to prove Theorem 1.2, we eed a Ramsey-theoretic result for touramets, which is iterestig o its ow right. A touramet T with a liear order < o its vertex set is called a ordered touramet ad is deoted by (T, <). A ordered touramet (T, <) is said to be a subtouramet of aother ordered touramet, (T, < ), if there is a fuctio f : V (T ) V (T ) satisfyig the coditios (i) f(u) < f(v) if ad oly if u < v, (ii) f(u)f(v) E(T ) if ad oly if uv E(T ). Theorem 1.3 For ay ordered touramet (T, <), there exists a touramet T such that, for every orderig < of T, (T, <) is a subtouramet ( of (T ), < ). Moreover, if T has vertices, there exists a T with the required property with O 3 log 2 vertices. We further show that the O( 3 log 2 ) estimate is ot very far from beig tight. I fact, if (T, <) is ay touramet o vertices ad T satisfies the coditio above for T, the T must have at least Ω( 2 ) vertices. The proof of the above theorem is very similar to the proof of the mai result of [RW89], which deals with a similar statemet for ordered iduced subgraphs. This ca be exteded to hypergraphs as well. By choosig a bigger touramet T, oe ca esure a sigle touramet that cotais all ordered touramets o vertices, i ay orderig. Specifically, we prove the followig. Theorem 1.4 Give a iteger N, let 0 be the largest iteger such that ( ) N 2 ( 0 2 ) 1, 0 ad put = 0 2. The, for all sufficietly large N, there exists a ordered touramet T o N vertices such that i ay orderig it cotais every ordered touramet o vertices. 3

4 Note that the above estimate for is clearly tight, up to a additive error of 2. A similar statemet holds for iduced subgraphs, as show i [BK93]. The rest of this paper is orgaized as follows. Theorem 1.1 is proved i Sectio 2. The proofs of Theorems 1.3 ad 1.4 appear i Sectio 3. Sectio 4 cotais the proof of Theorem Graphs with the Erdős-Hajal property I this sectio we prove Theorem 1.1. Obviously, it is sufficiet to show the followig weaker versio of the theorem. Theorem 2.1 Let H ad F be graphs havig the Erdős-Hajal property, V (H) = {v 1, v 2,..., v k }. The the graph H(F, v 2,..., v k ), obtaied by replacig v 1 with F, also has this property. Proof: Let H 0 deote the graph obtaied from H by the deletio of v 1. For simplicity, write H(F ) for H(F, v 2,..., v k ). Let G be a H(F )-free graph with vertices, ad assume that hom(g) < ε(h)δ. We would like to get a cotradictio, provided that δ > 0 is sufficietly small. Let m := δ > k. By the defiitio of ε(h), ay m-elemet subset of U V (G) must iduce at least oe subgraph isomorphic to H. Otherwise, we would fid a homogeeous subset of m ε(h) > hom(g) i the subgraph of G iduced by U, which is impossible. Therefore, G has at least ( ) ( m / k m k) iduced subgraphs isomorphic to H. For each of these subgraphs, fix a isomorphic embeddig of H ito G. Sice the umber of embeddigs of H 0 ito G is smaller tha ( 1) ( k + 2), there exists a embeddig, which ca be exteded to a embeddig of H i at least ( m) M := ( k ) (1) m k ( 1) ( k + 2) differet ways. I other words, there are k 1 vertices, v 2,..., v k V (G), ad there exists a at least M-elemet subset W V (G) such that, for every w W, f(v 1 ) = w, f(v i ) = v i (i = 2,..., k) is a isomorphic embeddig of H ito G. Cosider ow the subgraph G W of G iduced by W. This graph must be F -free, otherwise G would ot be H(F )-free. Sice F has the Erdős-Hajal property, we kow that hom(g W ) W ε(f ) M ε(f ). O the other had, ε(h)δ > hom(g) hom(g W ). 4

5 Comparig the last two iequalities ad pluggig i the value (1) for M, we obtai that ( δε(h)/ε(f ) > ( m) k + 1 k ) = m k ( 1) ( k + 2) m(m 1) (m k + 1) > 1 kδ, which gives the desired cotradictio, provided that δ < ε(f ) ε(h) + kε(f ). 3 Ramsey-type theorems for touramets The proof of Theorem 1.3 uses the probabilistic method. The basic idea is a slightly simplified versio of the mai argumet of Rödl ad Wikler i [RW89]. We eed the followig lemma. Lemma 3.1 Let t > > 1 be two positive itegers, ad let S = {a 1, a 2,..., a t } be a t-elemet set. Let g : S R = {1, 2,..., t} be a fuctio such that for every p R, we have {i : g(a i ) = p} =. Further, let f : S N = {1, 2,..., } be a radom fuctio obtaied by choosig, for each elemet a i S, radomly, idepedetly, ad with uiform distributio a value f(a i ) N. Let E be deote the evet that there exist 1 i 1 < i 2 <... < i t such that g(a ij ) g(a ik ) for all 1 j < k, ad f(a ij ) = j for all 1 j. The the probability that E does ot hold is at most 1 q=0 ( ) t q( 1) ( 1) t q q t q=0 ( ) 4et e t. Proof : To estimate the umber of fuctios f for which the evet E fails, we argue as follows. Give such a f, let i 1 be the smallest iteger (if it exists) such that f(a i1 ) = 1. Assumig i 1 < i 2 <... < i j 1 have already bee defied, ad assumig that f(i s ) = s for all s < j ad that the elemets g(a is ), s < j, are pairwise distict, let i j be the smallest iteger (if it exists) satisfyig i j > i j 1, f(a ij ) = j ad g(a ij ) g(a is ) for all s < j. Note that, sice the evet E fails, this process must termiate after some q 1 elemets i s have bee defied. Note also that if k is a idex satisfyig i s 1 < k < i s, ad g(a k ) differs from g(a ij ) for all j q (or eve just for all j < s), the f(a k ) caot be equal to s (sice otherwise we would have defied i s = k). Sice there is a similar restrictio for the value of f(a k ) for k < i 1 ad for k > i q, it follows that oce the sequece i 1 < i 2... < i q has bee defied, the value of f(a k ) ca attai at most 1 values for all but at most t q elemets a k. Therefore, the total umber of fuctios f for which the evet E fails is at most ( ) 1 t q( 1) ( 1) t q. q 5

6 Sice the total umber of possible fuctios f is t, the probability that E does ot hold is at most ( ) 1 t q( 1) ( 1) t q 1 ( ) et q ( ) q ( 1 q t q ( 1) q 1 1 ) t q=0 q=0 ( et (1 + 1 ) 1 ) e t ( ) 4et e t. Proof of Theorem 1.3: Let (T, <) be a ordered touramet o the set N = {1, 2,..., } of vertices, ordered aturally. We may ad will assume that is sufficietly large. Let c > 3 be a absolute costat, ad let t be the smallest iteger satisfyig t > c log such that t 1 is a prime. By the kow estimates for the distributio of primes, t = (1 + o(1))c log. Let P be a projective plae of order t 1. Each lie of P cotais precisely t poits, ad the umber of poits i P is (t 1) 2 + t < t 2. Replace each poit p P by a set S p of poits, where all sets S p are pairwise disjoit. Costruct a touramet T o the set p P S p of less tha t 2 vertices as follows. For every lie l i P, let f l : p l S p N = {1, 2,..., } be a radom fuctio, where each image f l (u) is chose radomly, uiformly ad idepedetly i N, ad the fuctios correspodig to differet lies are chose idepedetly. For u, v p l S p, where u S p, v S p ad p p, let uv be a directed edge if ad oly if f l (u)f l (v) is a directed edge of T. The edges with two edpoits i the same set S p are orieted arbitrarily. To complete the proof, we show that almost surely (that is, with probability tedig to 1 as teds to ifiity), T cotais a ordered copy of T i ay orderig. Fix a orderig < of T, ad let us estimate the probability that i this orderig (T, < ) cotais o ordered copy of T. For each lie l i the projective plae, the orderig < iduces a orderig of the t vertices p l S p. Let S = (a 1, a 2,..., a t ) be this iduced orderig. Defie g(a i ) = p if a i S p. The, for every p l, {i : g(a i ) = p} = S p =. Observe ow that, by Lemma 3.1, the probability that (T, <) is ot a subtouramet of the ordered subgraph of (T, < ) cosistig of all edges ruig betwee distict groups S p (p l), is at most ( 4et ) e t. This follows from the fact that, if the evet E i Lemma 3.1 holds for f = f l, the a i1,..., a i iduce a copy of T, as required. Sice the evets for distict lies are totally idepedet, the probability that (T, < ) cotais o ordered copy of (T, <) is at most ( ( 4et ) (t 1) 2 ) e t +t = e (1+o(1))c3 3 log 3. The total umber of orderigs of T is (((t 1) 2 + t))! e (1+o(1))3c2 3 log 3, ad as c > 3, by our choice, the probability that T fails to cotai a copy of T i some orderig is o(1), completig the proof. We ext show that the O( 3 log 2 ) upper boud caot be replaced by o( 2 ). We eed the followig well-kow result. 6

7 Lemma 3.2 ([D67], [A68]) The umber Aut(T ) of automorphisms of ay touramet T o vertices does ot exceed 3 ( 1)/2. Theorem 3.3 There exists a absolute costat b 1 3e 2 with the followig property. Let (T, <) be a ordered touramet o vertices, ad suppose T is aother touramet such that for every orderig < of T, (T, <) is a iduced subtouramet of T. The T has at least b 2 vertices. Proof: Let N be the umber of vertices of T. The the total umber of iduced labelled (but ot ecessarily ordered) copies of T i T is at most ( N ) Aut(T ), which, by Lemma 3.2, does ot exceed ( en ) 3 /2. It follows that the probability that for a radom orderig < of T, at least oe of these copies is ordered, is at most ( en ) 3 /2 1! ( 3e 2 N 2 ). If N < 2 /( 3e 2 ), this umber is less tha 1, implyig that there is a orderig < with o ordered copy of (T, <). Thus, we have N 2 /( 3e 2 ), completig the proof. The discussio for touramets ca be easily adapted to iduced subgraphs of graphs. A simple udirected graph H with a liear order < o its vertex set is called a ordered graph ad is deoted by (H, <). A ordered graph (H, <) is said to be a iduced subgraph of aother oe, (H, < ), if there is a fuctio f : V (H) V (H ) such that, for ay u, v V (H), (i) f(u) < f(v) if ad oly if u < v, (ii) f(u)f(v) E(H ) if ad oly if uv E(H). The proof of Theorem 1.3 ca be easily modified to deal with ordered graphs, givig the followig result of Rödl ad Wikler. Theorem 3.4 ([RW89]) For ay ordered graph (H, <), there exists a graph H such that, for every orderig < of H, (H, <) is a iduced subgraph ( of (H ), < ). Moreover, if H has vertices, there exists a H with the required property with O 3 log 2 vertices. Note that there is o otrivial aalogue of Theorem 3.3, sice the umber of automorphisms of a udirected graph o vertices ca be as large as!. I fact, if (H, <) is a ordered complete graph o vertices, the the graph H = H has oly vertices ad cotais a iduced ordered copy of (H, <) i ay orderig. Combiig the above argumets with some kow results about packigs, we ca exted the last result to iduced hypergraphs as well. Moreover, the estimate for hypergraphs with o edge of size less tha 3 is slightly better tha the correspodig result for graphs. A hypergraph H with a liear order < o its vertex set is called a ordered hypergraph ad is deoted by (H, <). A ordered hypergraph (H, <) is said to be a iduced subhypergraph of 7

8 aother oe, (H, < ), if there is a fuctio f : V (H) V (H ) such that, for ay u, v V (H), f(u) < f(v) if ad oly if u < v, ad a set of vertices forms a edge iff its image uder f forms a edge. Theorem 3.5 For ay ordered hypergraph (H, <) i which each edge cotais at least 3 vertices, there exists a hypergraph H such that, for every orderig < of H, (H, <) is a iduced subhypergraph of (H, < ). Moreover, if H has vertices, there exists a H with the required property with O ( 3) vertices. Proof: Let (H, <) be a ordered hypergraph o the set N = {1, 2,..., } of vertices, ordered aturally, where each edge of H is of size at least 3. Let c be a absolute costat such that 4ece c < 1/2 (c = 5, for example, will do). Let t be the smallest prime satisfyig t > c (the t = (1 + o(1))c.) As described i [K95], there is a simple, explicit costructio of a family L of t 3 subsets of a set P of size t 2 such that each member of l is of cardiality t ad the itersectio of o two members of L is of size more tha 2. Replace each elemet p P by a set S p of poits, where all sets S p are pairwise disjoit. Costruct a hypergraph H o the set p P S p of t 2 vertices as follows. For every l L, let f l : p l S p N = {1, 2,..., } be a radom fuctio, where each image f l (u) is chose, radomly, uiformly ad idepedetly i N, ad the fuctios correspodig to differet members l L are chose idepedetly. If u 1,..., u r are vertices i p l S p, the {u 1, u 2,..., u r } is a edge of H iff the vertices u i belog to pairwise distict sets S p, ad {f l (u 1 ), f l (u 2 ),..., f l (u r )} is a edge of H. Note that, sice the itersectio of ay two distict members of L is of size at most 2, ad H has o edges with fewer tha 3 vertices, oe of the edges defied above ca lie i the uio p l S p, for ay l L, l l. To complete the proof, we show that almost surely H cotais a ordered iduced copy of H i ay orderig. Fix a orderig < of H, ad let us estimate the probability that i this orderig (H, < ) cotais o ordered iduced copy of H. For each l L, the orderig < iduces a orderig of the t vertices p l S p. Let S = (a 1, a 2,..., a t ) be this iduced orderig. Defie g(a i ) = p if a i S p. The, for every p l, {i : g(a i ) = p} = S p =. Observe ow that, by Lemma 3.1, the probability that (H, <) is ot a iduced subhypergraph of the iduced ordered subhypergraph of (H, < ) o S, is at most ( 4et ) e t 2. This is true, because if the evet E i Lemma 3.1 holds for f = f l, the the vertices a i1,..., a i iduce a copy of T, as required. Sice the evets for distict sets l L are totally idepedet, the probability that (H, < ) cotais o ordered copy of (H, <) is at most (2 ) t3 = 2 (1+o(1))c3 4. The total umber of orderigs of H is (t 2 )! e (1+o(1))3c2 3 log, ad thus the probability that H fails to cotai a copy of H i some orderig is o(1), completig the proof. It is worth otig that the argumet i the proof of Theorem 3.3 also works for hypergraphs whose group of automorphisms is ot too large. I particular, if the hypergraph H i the statemet 8

9 of the last theorem has o otrivial automorphisms, the the umber of vertices of ay hypergraph H satisfyig the assertio of the theorem must be at least Ω( 2 ). Returig to touramets, we ow describe a proof of Theorem 1.4, usig Talagrad s Iequality [T95]. A alterative proof ca be give usig the methods of [BK93]. Proof of Theorem 1.4: Let N, 0 ad be as i the statemet of the theorem, ad let T be a radom touramet o the vertices 1, 2,..., N, obtaied by choosig, for each pair of vertices i, j of T, radomly, uiformly, ad idepedetly, either the edge ij or the edge ji. Wheever it is eeded, we assume that N is sufficietly large. To complete the proof, we show that almost surely i every orderig, T cotais a ordered copy of every touramet o vertices. To this ed, fix a orderig < of T, ad fix a ordered touramet T o vertices. We use Talagrad s Iequality (see, e.g., [AS00], Chapter 7) to estimate the probability that i this orderig T cotais o ordered copy of T. The computatio here is very similar to the oe estimatig the probability that the clique umber of the radom graph G(, 1/2) is less tha its expected value by at least 2. For each set K of vertices of T, let B K be the evet that the iduced subgraph of (T, < ) o K is a ordered copy of (T, <). The the probability Pr(B K ) of each evet B K is precisely 2 ( 2). Defie µ = ( N ) 2 ( 2), ad ote that this is the expected umber of ordered copies of (T, <) i (T, < ). A simple computatio shows that the umber 0 defied i the statemet of the theorem satisfies 0 = (1 + o(1))2 log 2 N, implyig that for the fuctio f(m) = ( N m) 2 ( m 2 ) ad for every m close to 0, we have f(m + 1)/f(m) = N 1+o(1). Therefore, µ N 2 o(1). For two subsets K ad K, each cotaiig vertices of T, let K K stad for the fact that 2 K K 1. Defie, further, = K K Pr(B K B K ), where the sum rages over all ordered pairs (K, K ) with K K. Therefore, 1 2 is the expected umber of pairs of ordered copies of T that share a edge. The techical part of the proof is a careful estimate of the quatity /µ 2. Observe that if K K = i ( 2) the Pr(B K B K ) 2 2( 2)+( i 2). I fact, Pr(B K B K ) is equal either to zero, or to the right-had side of the above expressio. Thus, it follows that 1 i=2 i, where i = ( N )( i )( N i ) 2 2( 2)+( i 2). Therefore, ( N )( N ) i µ 2 = i)( i 2 2( 2)+( 2) i ) 22 2( 2) ( N 9

10 It follows that ( N ) = i)( i 2 ( 2) i ( N ) ( ( ) ( ) 2 i N )i 2 (i 2) i N 2(i 1)/2. ad that for each i satisfyig, say, 3 i 100, we have Furthermore, for 100 < i 1.9 log 2 N, we have 2 µ N 2, (2) i 6 = O( ). (3) µ 2 N 3 i µ 2 ( 2 N 0.05 )i < 1 N 5. (4) For every i i the rage 1.9 log 2 N i 1, put i = j, ad ote that 1 j (0.1+o(1)) log 2 N ad ( N )( N ) i µ 2 1 i N 2 o(1) µ = 1 j)( j 2 ( 2) j( j) ( 2) j N 2 o(1) ( N ) 2 ( 2) ( )( ) 1 N N 2 o(1) 2 j( j) 1 ( j j N 2 o(1) N2 ( j)) j ( ) 1 j 1 N 2 o(1) N 0.9 o(1) N 2.9 o(1). Combiig the last iequality with iequalities (2), (3) ad (4), we coclude that (2+o(1)) 4. µ 2 N 2 Let X = h(t ) deote the maximum umber of pairwise edge-disjoit ordered copies of T i (T, < ). We claim that the expected value of X = h(t ) satisfies E(X) ( o(1))n2 4. (5) To see this, defie p = N 2, ad ote that, by a simple computatio, we have p < 1. Let S be 2 4 µ a radom collectio of ordered copies of T i T obtaied by choosig each ordered copy of T i T to be a member of S, radomly ad idepedetly, with probability p. The expected umber of copies of T i S is pµ, ad the expected umber of pairs of members of S that share a edge is 1 2 p2. By omittig a arbitrarily chose member of each such pair, we obtai a collectio of pairwise edge-disjoit copies of T whose expected umber is at least pµ 1 2 p2. Thus E(X) pµ p2 2 N N 4 4 (2 + o(1)) 88 N 2 = (1 4 + o(1))n2 4, 10

11 establishig (5). To apply Talagrad s Iequality (i the form preseted, for example, i [AS00], Chapter 7), ote that h(t ) is a Lipschitz fuctio, that is h(t ) h(t ) 1 if T, T differ i the orietatio of at most oe edge. Note also that h is f-certifiable for f(s) = ( 2) s. That is, wheever h(t ) s there is a set of at most ( 2) s orieted edges of T such that for every ordered touramet T which agrees with T o these edges, we have h(t ) s. By Talalgrad s Iequality we coclude that for every b ad t P r[x b t f(b)] P r[x b] e t2 /4. (6) Let B deote the media of X = h(t ). Without tryig to optimize the absolute costats, we claim that B N (7) Ideed, assume this is false, ad apply (6) with b = N obtai that P r[x N N ] P r[x N 2 ad t = N. As f(b) = ( ] e N 2 /(64 6). ) N 2 N , we Sice, by assumptio, B < N , the first term of the left-had side is at least 1/2, ad we coclude that P r[x N ] 2e N 2 /(64 6). As X = h(t ) ( N ) for every T, this implies that E(X) N 2 ( ) N e N 2 /(64 6) = N o(1), cotradictig (5) ad hece provig (7). We ca ow apply (6) with b = N ad t = N 4 3 to obtai that P r[x = 0]P r[x b] P r[x b t f(b)]p r[x b] e N 2 /(64 6). By (7), we have P r[x b] 1/2, ad hece P r[x = 0] 2e N 2 /(64 6). Thus, we have proved that, for every fixed orderig of T ad for every fixed ordered T, the probability that T cotais o ordered copy of T is at most 2e N 2 /(64 6). Sice the total umber of orderigs of T is less tha N N = e N log N ad the total umber of touramets T o vertices is 2 ( 2) we coclude that the probability that (T, < ) fails to cotai some touramet of size i some orderig is at most e N log N 2 ( 2) 2e N 2 /(64 6) = o(1). 11

12 This completes the proof. The above proof ca be modified to deal with graphs i the place of touramets. We obtai the followig, which is a very slight umerical improvemet of the mai result i [BK93]. Theorem 3.6 (see also [BK93]) Give a iteger N, let 0 be the largest iteger such that ( ) N 2 ( 0 2 ) 1, 0 ad put = 0 2. The, for all sufficietly large N, the followig holds almost surely. The radom graph G(N, 1/2) cotais, i ay orderig, a iduced copy of every ordered graph o at most vertices. 4 Touramets ad H-free graphs I this sectio, we prove Theorem 1.2. We eed the followig wellkow Lemma 4.1 ([ES35]) For ay two total orderigs of the same (k 2 + 1)-elemet set V, there is a (k + 1)-elemet subset U V such that either the order of ay two elemets of U is the same, or the order of ay two elemets is opposite i the two orderigs. We say that a touramet T has the Erdős-Hajal property if there exists a positive ε = ε(t ) such that every T -free touramet with vertices has a trasitive subtouramet whose size is at least ε. To ay touramet T ad to ay orderig < of its vertex set, assig a ordered graph (H(T ), <) o the same vertex set, as follows. Joi two vertices u < v by a edge of H(T ) if ad oly if the edge coectig them i T was directed towards v. Similarly, assig to ay ordered graph (H, <) a ordered touramet (T (H), <) with the same vertex set, by coectig u < v with a edge directed towards v if uv E(H) ad with a edge directed towards u if uv / E(H). Now we have everythig eeded for the Proof of Theorem 1.2: Assume first that Cojecture 1 is true, i.e., every graph has the Erdős- Hajal property. Let T be a touramet. We wat to show that T also has the Erdős-Hajal property. Choose a arbitrary orderig < of the vertex set of T. Applyig Theorem 3.4 to the ordered graph (H(T ), <) associated with T ad <, we obtai that there exists a graph H with the property that, for ay orderig < of H, (H(T ), <) is a iduced subgraph of (H, < ). By Cojecture 1, there exists a ε(h ) > 0 such that every H -free graph with vertices has a homogeeous subset of size at least ε(h ). 12

13 Cosider ow a T -free touramet T with vertices ad a orderig < of V (T ). The the ordered graph (H(T ), < ) associated with them caot cotai a iduced subgraph isomorphic to H (because, o matter how it is ordered, this would yield a copy of T i T ). Thus, H(T ) must have a homogeeous set of size at least ε(h ). However, a homogeeous set i (H(T ), < ) correspods to a trasitive subtouramet i T. The proof of the reverse statemet is very similar, but the roles of graphs ad touramets have to be switched. Assume that Cojecture 2 is true, ad let H be a arbitrary graph. To establish that H has the Erdős-Hajal property, fix a liear order < o V (H), ad deote the associated ordered touramet by (T (H), <). By Theorem 1.3, there exists a touramet T with the property that, for ay orderig < of T, (T (H), <) is a subtouramet of (T, < ). By Cojecture 2, there exists a ε(t ) > 0 such that every T -free touramet with vertices has a trasitive subtouramet of size at least ε(t ). Cosider ow a H-free graph H with vertices ad a orderig < of V (H ). The the ordered touramet (T (H ), < ) associated with them caot cotai a subtouramet isomorphic to T (because, o matter how it is ordered, this would yield a copy of H i H ). Thus, T (H ) must have a trasitive subtouramet of size at least ε(h ). However, by Lemma 4.1, ay such subtouramet has at least ε(h )/2 vertices such that, with respect to the orderig <, either all edges coectig them are directed towards their larger edpoits, or all of them are directed towards their smaller edpoits. These vertices iduce a complete or a empty subgraph of H, respectively. Ackowledgmet We would like to thak two aoymous referees for helpful commets. Refereces [AS00] N. Alo ad J. H. Specer: The Probabilistic Method, Secod Editio, Wiley, [A68] B. Alspach: A combiatorial proof of a cojecture of Goldberg ad Moo, Caad. Math. Bull. 11 (1968), [BK93] G. Brightwell ad Y. Kohayakawa: Ramsey properties of orietatios of graphs, Radom Structures ad Algorithms 4 (1993), [D67] J. D. Dixo: The maximum order of the group of a touramet, Caad. Math. Bull. 10 (1967), [EEH73] R. C. Etriger, P. Erdős, ad C. C. Harer: Some extremal properties cocerig trasitivity i graphs, Periodica Mathematica Hugarica 3 (1973), [E47] P. Erdős: Some remarks o the theory of graphs, Bulleti of the America Mathematical Society 53 (1947),

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