Ramseytype theorems with forbidden subgraphs


 Tiffany Reynolds
 4 years ago
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1 Ramseytype theorems with forbidde subgraphs Noga Alo Jáos Pach József Solymosi Abstract A graph is called Hfree 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 Hfree 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 Hfree 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, TelAviv Uiversity, Tel Aviv, Israel. 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 H1364 Budapest, P.O.B Supported by NSF grat CR , PSCCUNY Research Award , ad OTKAT Computer ad Automatio Istitute of the Hugaria Academy of Scieces H1518 Budapest, P.O.B. 63. Supported by TKI grat ad OTKAT
2 Cojecture 1 For every graph H, there exists a positive ε = ε(h) such that every Hfree 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 vertexdisjoit 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 ErdősHajal 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 ErdősHajal property, the so does H(F 1,..., F k ). I other words, the ErdősHajal 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 ErdősHajal property No operfect graph is kow to have the ErdősHajal 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 operfect graph, the cycle of legth 5, has the ErdősHajal 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 Ramseytheoretic 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 ErdősHajal 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 ErdősHajal 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 melemet 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 Melemet 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 ErdősHajal 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 Ramseytype 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 telemet 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 wellkow 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 righthad 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 edgedisjoit 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 edgedisjoit 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 fcertifiable 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 lefthad 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 Hfree 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 ErdősHajal 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 ErdősHajal 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 ErdősHajal 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 Hfree 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),
14 [EH89] P. Erdős ad A. Hajal: Ramseytype theorems, Discrete Mathematics 25 (1989), [ES35] P. Erdős ad G. Szekeres: A combiatorial problem i geometry, Compositio Math. 2 (1935), [G97] A. Gyárfás: Reflectios o a problem of Erdős ad Hajal, i: The Mathematics of Paul Erdős (R. L. Graham ad J. Nešetřil, eds.), Algorithms ad Combiatorics 14, Volume II, SprigerVerlag, Heidelberg, 1997, [K95] N. N. Kuzjuri: O the differece betwee asymptotically good packigs ad coverigs, Europea J. Comb. 16 (1995), [L72] L. Lovász: Normal hypergraphs ad the perfect graph cojecture, Discrete Mathematics 2 (1972), [L83] L. Lovász: Perfect Graphs, i: Selected Topics i Graph Theory, Volume 2 (L. W. Beieke, R. J. Wilso, eds.), Academic Prcess, LodoNew York, 1983, [RW89] V. Rödl ad P. Wikler: A Ramseytype theorem for orderigs of a graph, SIAM Joural of Discrete Mathematics 2 (1989), [Se74] D. Seische: O a property of the class of colorable graphs, Joural of Combiatorial Theory, Ser. B 16 (1974), [S74] J. Specer: Radom regular touramets, Periodica Mathematica Hugarica 5 (1974), [T95] M. Talagrad, Cocetratio of measure ad isoperimetric iequalities i product spaces, Ist. Hautes Études Sci. Publ. Math. 81 (1995)
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