Ramsey numbers of sparse hypergraphs

Transcription

1 Ramsey numbers of sparse hypergraphs David Conon Jacob Fox Benny Sudakov Abstract We give a short proof that any k-uniform hypergraph H on n vertices with bounded degree has Ramsey number at most c(, k)n, for an appropriate constant c(, k). This resut was recenty proved by severa authors, but those proofs are a based on appications of the hypergraph reguarity method. Here we give a much simper, sef-contained proof which uses new techniques deveoped recenty by the authors together with an argument of Kostochka and Röd. Moreover, our method demonstrates that, for k 4, c(, k) where the tower is of height k and the constant c depends on k. It significanty improves on the Ackermann-type upper bound that arises from the reguarity proofs, and we present a construction which shows that, at east in certain cases, this bound is not far from best possibe. Our methods aso aow us to prove quite sharp resuts on the Ramsey number of hypergraphs with at most m edges. c, 1 Introduction For a graph H, the Ramsey number r(h) is the east positive integer N such that, in every twocoouring of the edges of compete graph K N on N vertices, there is a monochromatic copy of H. Ramsey s theorem states that r(h) exists for every graph H. A cassica resut of Erdős and Szekeres, which is a quantitative version of Ramsey s theorem, impies that r(k k ) 2 2k for every positive integer k. Erdős showed using probabiistic arguments that r(k k ) > 2 k/2 for k > 2. Over the ast sixty years, there have been severa improvements on these bounds (see, e.g., [4]). However, despite efforts by various researchers, the constant factors in the above exponents remain the same. Determining or estimating Ramsey numbers is one of the centra probem in combinatorics, see the book Ramsey theory [17] for detais. Besides the compete graph, the next most cassica topic in this area concerns the Ramsey numbers of sparse graphs, i.e., graphs with certain upper bound constraints on the degrees of the vertices. The study of these Ramsey numbers was initiated by Burr and Erdős in 1975, and this topic has since paced a centra roe in graph Ramsey theory. Burr and Erdős conjectured, and it was proved by Chváta, Röd, Szemerédi and Trotter [3], that for every graph G on n vertices and maximum degree, r(g) c( )n. St John s Coege, Cambridge, United Kingdom. E-mai: D.Conon@dpmms.cam.ac.uk Department of Mathematics, Princeton, Princeton, NJ. Emai: jacobfox@math.princeton.edu. Research supported by an NSF Graduate Research Feowship and a Princeton Centennia Feowship. Department of Mathematics, UCLA, Los Angees, CA and Institute for Advanced Study, Princeton, NJ. Emai: bsudakov@math.uca.edu. Research supported in part by NSF CAREER award DMS , NSF grants DMS and DMS , by a USA-Israei BSF grant, and by the State of New Jersey. 1

2 Their proof of this theorem is a cassic appication of Szemerédi s beautifu reguarity emma. However, the use of this emma makes the upper bound on c( ) grow as a tower of 2s with height proportiona to. Eaton [8] used a variant of the reguarity emma to obtain the upper bound c( ) 2 2c for some fixed c. A nove approach of Graham, Röd, Rucinski [15] that did not use any form of the reguarity emma gives the upper bound c( ) 2 c og2 for some fixed c. In the other direction, in [16] they proved that there is a positive constant c such that, for every 2 and n + 1, there is a bipartite graph G with n vertices and maximum degree at most satisfying r(g) 2 c n. Recenty, the authors [5], [12] cosed the gap for bipartite graphs by showing that, for every bipartite graph G with n vertices and maximum degree, r(g) 2 c n for some fixed c. A hypergraph H = (V, E) consists of a vertex set V and an edge set E, which is a coection of subsets of V. A hypergraph is k-uniform if each edge has exacty k vertices. The Ramsey number r(h) of a k-uniform hypergraph H is the smaest number N such that, in any 2-coouring of the edges of the compete k-uniform hypergraph K (k) N, there is guaranteed to be a monochromatic copy of H. The existence of these numbers was proven by Ramsey [24], but no proper consideration of the vaues of these numbers was made unti the fifties, when Erdős and Rado [11]. To understand the growth of Ramsey numbers for hypergraphs, it is usefu to introduce the tower function t i (x), which is defined by t 1 (x) = x and t i+1 (x) = 2 ti(x), i.e., t i+1 (x) = where the number of 2s in the tower is i. Erdős and Rado showed that for H being the compete k-uniform hypergraph K (k), r(h) t k (c), where the constant c depends on k. In the other direction, Erdős and Hajna (see [17]) proved that for H = K (k), r(h) t k 1 (c 2 ), where the constant c depends on k. One can naturay try to extend the sparse graph Ramsey resuts to hypergraphs. Kostochka and Röd [20] showed that for every ɛ > 0, the Ramsey number of any k-uniform hypergraph H with n vertices and maximum degree satisfies r(h) c(, k, ɛ)n 1+ɛ, where c(, k, ɛ) ony depends on, k, and ɛ. Since the first proof of the sparse graph Ramsey theorem used Szemerédi s reguarity emma, it was therefore natura to expect that, given the recent advances in deveoping a hypergraph reguarity method [14, 25, 23], inear bounds might as we be provabe for hypergraphs. Such a program was indeed recenty pursued by severa authors (Cooey, Fountouakis, Kühn, and Osthus [6, 7]; Nage, Osen, Röd, and Schacht [22]; Ishigami [18]), with the resut that we now have the foowing theorem: Theorem 1 Let and k be positive integers. Then there exists a constant c(, k) such that the Ramsey number of any k-uniform hypergraph H with n vertices and maximum degree satisfies x r(h) c(, k)n. In this paper we wi give a short proof of this theorem, which is much simper and avoids a use of the reguarity emma, buiding instead on techniques deveoped recenty by Conon [5] and by Fox and Sudakov [12] in order to study embeddings of sparse bipartite graphs in dense graphs., 2

3 The first main resut of this paper is an extension of this work from graphs to hypergraphs. An - uniform hypergraph is -partite if there is a partition of the vertex set into parts such that each edge has exacty one vertex in each part. We prove the foowing Turán-type resut for -uniform -partite hypergraphs: Theorem 2 There exists a constant c = c() such that if F is an -uniform -partite hypergraph with n vertices and maximum degree and G is an -uniform -partite hypergraph with parts of size N (ɛ/2) c 1 n and at east ɛn edges, then G contains a copy of F. Then, in section 3, we wi prove Theorem 1 by appying an argument of Kostochka and Röd which shows that the Ramsey probem for genera hypergraphs may be reduced to an appication of the Turán theorem in the -uniform -partite case. This argument combined with our Theorem 2 shows that, for k 4 and k-uniform hypergraph H with n vertices and maximum degree, r(h) t k (c )n, where the constant c depends on k. For k = 3, the proof shows that r(h) t 3 (c og )n. This is ceary much better than the Ackermann-type upper bound that arises from the reguarity proofs. The tower-type upper bound cannot be avoided as demonstrated by the ower bound of Erdős and Hajna for the Ramsey number of the compete k-uniform hypergraph on n vertices. This hypergraph has maximum degree = ( ) n 1 k 1 and Ramsey number at east tk 1 (c 2 k 1 )n, where the constant c depends on k. For k-uniform hypergraphs H 1,..., H q, the muticoour Ramsey number r(h 1,..., H q ) is the minimum N such that, in any q-coouring of the edges of the compete k-uniform hypergraph K (k) N with coours 1,..., q, there is a monochromatic copy of H i in coour i for some i, 1 i q. The proof of Theorem 1 presented here extends in a straightforward manner to the muticoour generaisation, which states that for a positive integers, k, and q, there exists a constant c(, k, q) such that, if H 1,..., H q are k-uniform hypergraphs each with n vertices and maximum degree, then r(h 1,..., H q ) c(, k, q)n. The proof demonstrates that may take c(, k, q) t k (c ) for k 4 and c(, 3, q) t 3 (c og ), where the constant c depends on k and q. In the other direction, in Section 4 we construct, for each sufficienty arge, a 3-uniform hypergraph H with maximum degree at most for which the 4-coour Ramsey number of H satisfies r(h, H, H, H) t 3 (c )n, where n is the number of vertices of H. This exampe shows that our upper bound for hypergraph Ramsey numbers is probaby cose to being best possibe. The same exampe aso shows that there is a 3-uniform hypergraph H with m edges for which the 4-coour Ramsey number of H is at east t 3 (c m). On the other hand, one can easiy deduce from the proof of Theorem 1 that for any k-uniform hypegraph H with m edges, we have that the q-coour Ramsey number of H satisfies r(h,, H) t 3 (c m og m) for k = 3, and r(h,, H) t k (c m) for k 4, where c depends on k and q. 2 A Turán theorem for -uniform -partite hypergraphs The foowing is a generaisation to hypergraphs of a emma which has appeared increasingy in the iterature on Ramsey theory, whose proof uses a probabiistic argument known as dependent random choice. Eary versions of this technique were deveoped in the papers [13], [19], [26]. Later, variants 3

4 were discovered and appied to various Ramsey and density-type probems (see, e.g., [21, 1, 27, 20, 12, 5]). We define the weight w(s) of a set S of edges in a hypergraph to be the size of the union of these edges. Lemma 1 Suppose s, are positive integers, ɛ, β > 0, and G r = (V 1,, V r ; E) is an r-uniform r-partite hypergraph with V 1 = V 2 = = V r = N and at east ɛn r edges. Then there exists an (r 1)-uniform (r 1)-partite hypergraph G r 1 on the vertex sets V 2,, V r which has at east ɛs 2 N r 1 edges and such that for each nonnegative integer w (r 1), there are at most 4r ɛ s β s w r r w N w dangerous sets of edges of G r 1 with weight w, where a set S of edges of G r 1 is dangerous if S and the number of vertices v V 1 such that for every edge e S, e + v G r is ess than βn. Proof: Let C be the compete (r 1)-uniform (r 1)-partite hypergraph on the vertex sets V 2,, V r. For any edge e in C, et d(e) be the degree of e in G r, i.e., the number of vertices in V 1 such that e + v G r. Let T be a set of s random vertices of V 1, chosen uniformy with repetitions. Let A be the set of edges in C which are common neighbours of the vertices of T, i.e., an edge e of C is in A if e + v is an edge of G r for a v T. Let X denote the cardinaity of A. We wi show that with positive probabiity, the set A wi be the edge set of a hypergraph G r 1 on vertex sets V 2,..., V r with the desired properties. By inearity of expectation and by convexity of f(z) = z s, ( ) d(e) s E[X] = P[e I] = e C e C ( N r 1 e C d(e) N r 1 N s ) s N N r 1 (ɛn) s N s = ɛ s N r 1. Note that X N r 1 since C has N r 1 edges. Letting p denote the probabiity that X E[X]/2, we have E[X] (1 p)e[x]/2 + pn r 1 E[X]/2 + pn r 1. So the probabiity p that X E[X]/2 ɛ s N r 1 /2 satisfies p E[X] ɛ s /2. 2N r 1 The number of subsets S of V 2... V r of size w is ( ) (r 1)N w. For a given w-set S, the number of coections {e 1,..., e t } of size t with e i = r 1, and e i S for 1 i t is ( ( r 1) w ) t. Hence, summing over a nonnegative t, the number of sets of edges of C with weight w and size at most is at most t=0 (( w r 1 ))( ) (r 1)N w r (rn) w = w r r w N w, t w Let Y w denote the random variabe counting the number of dangerous sets S of edges of G r 1 with weight w. We next give an upper bound on ) E[Y w ]. For a given set S of edges of C, the probabiity s, that S is a subset of edges of G r 1 is where N(S) denotes the set of vertices v V1 with ( N(S) N v + e an edge of G r for a e S. So if S satisfies N(S) < βn, then the probabiity that S is a subset of edges of G r 1 is ess than β s. By inearity of expectation, we have E[Y w ] < β s w r r w N w. Let α = 4r ɛ s. Since Y w is a nonnegative random variabe, Markov s inequaity impies that P (Y w αe[y w ]) 1 α. Hence, the probabiity that there is a nonnegative integer w (r 1) 4

5 with Y w αβ s w r r w N w is at most r /α = ɛ s /4. Since the probabiity that X ɛs 2 N r 1 is at east ɛ s /2, we can satisfy the conditions of the emma with probabiity at east ɛ s /4. By simpy iterating the previous emma 1 times, we obtain the foowing coroary. Coroary 1 Suppose s, are positive integers, ɛ, β > 0, and G = (V 1,, V ; E ) is an -uniform -partite hypergraph with V 1 = V 2 = = V = N and at east ɛn edges. Let δ = ɛ and δ r 1 = δr/2 s for 2 r. Then, for 1 r 1, there are r-uniform r-partite hypergraphs G r = (V r+1,..., V, E r ) with the foowing properties: 1. G r has at east δ r N r edges for 1 r, and 2. for 2 r and each nonnegative integer w (r 1), there are at most 4r δr s β s w r r w N w dangerous sets of G r 1 with weight w, where a set S of edges of G r 1 is dangerous if S and the number of vertices v V r+1 such that for every edge e S, e + v G r is ess than βn. This is a the preparation we need before proving our main contribution, Theorem 2. For the proof, we wi use Coroary 1 and then show how to embed F into G. The atter part is cosey reated to the many embedding resuts proven by Fox and Sudakov in [12]. We wi actuay prove the foowing more precise version of Theorem 2: Theorem 3 Let 3, F be an -uniform -partite hypergraph, on vertex sets W 1,, W, with at most n vertices and maximum degree. Let G be an -uniform -partite graph, on vertex sets V 1,, V with V 1 = = V = N, with at east ɛn edges. Then, provided that N (ɛ/2) (2 ) 1 n, G contains a copy of F. Proof: We appy Coroary 1 with s = 2, δ = ɛ, δ i 1 = δi s /2 for 2 i, and β = 2 (ɛ/2)(2 ) 1 to get hypergraphs G 1,..., G 1. It is easy to check by induction on i that δ i = 2 (si 1)/(s 1) ɛ si, so and δ 1 N βn 2n. δ 1 = 2 (s 1 1)/(s 1) ɛ s 1 2 (ɛ/2) (2 ) 1 = β We now construct an -unform bad hypergraph B with vertex set V 1... V where each edge of B has exacty vertices in each V i. A set T V 1... V which contains exacty vertices in each V i is an edge of B if and ony if there is a dangerous set S of edges of G r for some r, 1 r 1, such that the union of the edges of S is a subset of T. In other words, an edge of B is just an extension of the union of the edges of a dangerous set. For a particuar dangerous set S of edges with weight w in some G r, the number of edges of B that are extensions of the union of the edges in S is at most N w since there are at most N ways to pick each of the w remaining vertices that make up an edge. Summing over a r and w, and using the fact that 3 and δ s 2 = 2δ 1 2β, the number of edges of B is at most r=2 ( 1) w=0 4r δ s r β s w r r w N 2 2 β s 1 ( ) N = ( 2 ) β 1 β N 2 1+3( )2 β 1 β N 2 1+3( )2 2 (1 (2 ) 1)( 1) β N ( ) β ( ) N 2 4( )2 β N. 4 5

6 Ca a set U V 1... V with at most vertices in each V i bad if there are at east ( ) β U ( ) N 4 U edges of B that contain U; otherwise ca U good. Note that the above cacuation on the number of edges of B demonstrates that the empty set is good. We next prove the foowing important caim. Caim 1 If S is a dangerous set of edges in G r for some r, 1 r 1, and U is a good set, then the union of the edges in S is not a subset of U. Proof: Suppose for contradiction that the union of the edges in S is a subset of U. The number of extensions of U to a set which contains exacty vertices in each V i is i=1 ( ) N Vi U V i U since we can pick for each i any V i U vertices of V i \ U to extend U. By definition, a of these sets are edges in B. Using the simpe fact that if x 1,..., x are nonnegative integers then i=1 x i! ( i=1 x i)!, it is straightforward to check that i=1 ( ) N Vi U V i U (N/2) U ( V i U )! 1 i=1 ( ) β U ( ) N, 4 U ( ) 1 U ( ) N 2 U which contradicts U being good. Given a good set U with V i U < and v V i \ U, we say v is bad with respect to U if U {v} is bad. Let B U denote the set of vertices that are bad with respect to U. We wi show that for U good we have B U βn 4. Indeed, suppose B U > βn 4. Then the number of edges of B containing U is at east ( ) B U β U 1 ( ) ( ) N β U ( ) N >, U 4 U 1 4 U contradicting the fact that U is good. Fix a abeing {v 1,, v n } of the vertices of F such that a vertices in W i+1 precede a those in W i for a i = 1,, 1. For each i, et L i = {v 1,, v i }. For each vertex v h, the trace neighbourhood N(v h ) is the set of vertices v m with m < h that are in an edge of F with v m. Note that N(v h ) contains at most vertices in each W r since F has maximum degree. We wi find an embedding f of the vertices of F such that f(w r ) V r for 1 r and for each i N, 1. f(n(v) L i ) is good for each vertex v of F, and 2. f(e L i ) is an edge of G r for each edge e of F, where r = e L i. 6

7 The proof wi be compete once we find such an embedding f since, for each edge e of F, f(e L n ) = f(e) is an edge of G, so f provides an embedding of F in G. The embedding wi be constructed one vertex at a time, in increasing order of subscript, so the proof wi be by induction on i. As noted earier, the empty set is good, so our base case i = 0 is satisfied. Suppose then that at step i, we have found an embedding f of v 1,..., v i such that 1. for each vertex v of F, f(n(v) L i ) is good, and 2. for each edge e of F, f(e L i ) is an edge of G r, where r = e L i. Let j be such that v i+1 W j. Let e 1,..., e d denote the edges of F that contain v i+1 and e 1,..., e d denote the truncations of e 1,..., e d by deeting a j vertices from each e t that are in some W h with h j. Each e t consists of one vertex from each W h with h > j. Aso, d since F has maximum degree. Since F has maximum degree, there are ess than vertices v for which v i+1 N(v). For each β such v, f(n(v) L i ) is good, so there are at most 4 N vertices w in V j for which f(n(v) L i ) w is bad. Adding over a such v, we concude that there are at most β 4 N bad vertices in a associated with v i+1. Suppose we are sti embedding vertices of W in V. Since the edge set of G 1 is just a subset of V whose size by Coroary 1 is at east δ 1 N = βn, then we can choose any of these at east βn vertices other than f(v 1 ),..., f(v i ) for f(v i+1 ) to satisfy the second of the two desired properties for f(v i+1 ). We see that there are at east βn i β 4 N > 3βN 4 n > 0 vertices to choose from for f(v i+1 ) to satisfy both of the desired properties. If, now, we have chosen a of the vertices in W,, W j+1 and we are trying to embed vertex v i+1 in W j (we may have aready embedded other vertices in W j ), we can do so. To see this, by the induction hypothesis, f(n(v i+1 ) L i ) = f(n(v i+1 )) = d t=1 f(e t) is good. By Caim 1, this impies that the set {f(e 1 ),..., f(e d )} of edges of G j is not bad, i.e., there are at east βn vertices v V j such that f(e t) v is an edge of G j+1 for 1 t d. Therefore, since there are at most β 4 N bad vertices associated with v i+1 and we have aready chosen f(v 1 ),..., f(v i ), we have at east 3 4 βn i > 3 4 βn n > 0 choices for f(v i+1), which competes the proof. 3 The Ramsey theorem We are now ready to prove Theorem 1 in the foowing form: Theorem 4 Let and k 3 be positive integers. Then the Ramsey number of any k-uniform hypergraph H with n vertices and maximum degree satisfies where r k () = r(k (k) ). r(h) r k (k ) (2k 2 ) k n, Proof: We use the argument of Kostochka and Röd [20] together with Theorem 3. Let = (k 1) + 1. Suppose we have a red-bue coouring of the compete k-uniform hypergraph on N vertices. 7

8 Let G be the hypergraph consisting of a the red edges and et r k () be the Ramsey number of the hypergraph K (k). Then, in each subset of the vertices of size r k (), there is at east one monochromatic K (k). Counting over a such sets and dividing out by possibe mutipe counts we see that we have at east ( N ) r k () ) N r k () ( N r k () monochromatic K (k). Therefore, either G or its compement G contains at east N /2r k () ciques K (k). We wi suppose that it is G. Now we pass instead to considering the -uniform hypergraph G (), the edges of which are exacty those -tupes which form compete K (k) in G. This hypergraph has at east N /2r k () edges. Partition its N! vertex set randomy into parts V 1,, V of equa size N/. The tota number of partitions is (N )! (N/)! and, for any given edge e, there are! partitions such that each vertex of this edge is in a (N/ 1)! different part of the partition. Therefore, the expected number of edges with one vertex in each set of the random partition is at east ( / ) e(g () (N )! N! )! (N/ 1)! (N/)! N 2r k ()! (N )! (N/)! (N/ 1)! N! N! 2r k () =! ( ) N 2r k (). Now choose such a partition and et Ĝ() be the -uniform -partite subhypergraph of G () consisting of those edges of G () which have one edge in each partite set. Note that Ĝ() has N/ vertices in each part and at east ɛ ( ) N edges, where ɛ =!. 2r k () Now we extend hypergraph H to an -uniform -partite hypergraph H (). We first note that the vertices of H can be partitioned into subsets A 1,..., A such that each edge of H has at most one vertex in each part. This is equivaent to saying that the graph H with the same vertex set as H and with two edges adjacent if they ie in an edge of H has chromatic number at most. Since H has maximum degree at most (k 1), it has chromatic number at most (k 1) + 1 =. For each edge e of H, we add one auxiiary vertex to each A i which is disjoint from e (in tota k vertices). Note that the maximum degree of H () remains. The tota number of auxiiary vertices added is at most n k ( k) < ( 1)n since there are k auxiiary vertices for each edge and the tota number of edges of H is at most n k. Hence, H() has ess than 2 n vertices. Appying Theorem 3 with F = H (), G = Ĝ(), and ɛ =! we see that, provided 2r k () N (ɛ/2) (2 ) 1 2 n, Ĝ () contains a copy of H (). But now, by the construction of H (), this impies that every edge in H is contained inside an edge of Ĝ(). But Ĝ() was chosen in such a way that every k-tupe within any edge of Ĝ() is an edge in G. Therefore G contains a copy of H, so we are done. As mentioned in the introduction, the proof of Theorem 1 presented here extends in a straightforward manner to the foowing muticoour generaisation. Theorem 5 For a positive integers, k, and q, there exists a constant c(, k, q) such that, if H 1,..., H q are k-uniform hypergraphs each with n vertices and maximum degree, then r(h 1,..., H q ) c(, k, q)n. 8

9 The ony difference in the proof is in Theorem 4, where we repace r k () by r k (; q), the q-coour Ramsey number for the compete k-uniform hypergraph on vertices. Erdős and Rado [11] showed that r k (; q) t k (c), where the constant c depends on k and q. We therefore may take c(, k, q) t k (c ) for k 4 and c(, 3, q) t 3 (c og ), where the constant c depends on k and q. Remark: The strong chromatic number of a hypergraph H is the minimum number of coors required to coour the vertices of H so that each edge of H has no repeated coour. The proof of Theorem 4 demonstrates that if H is a k-uniform hypergraph with n vertices, maximum degree, and strong chromatic number, then the q-coour Ramsey number of H satisfies r(h,, H) r k (; q) (2 ). Indeed, in the proof of Theorem 4, we ony used the fact that the vertices of H can be partitioned into parts such that every edge has at most one vertex in each part. 4 Lower bound construction The foowing theorem demonstrates that our upper bound for hypergraph Ramsey numbers proved in the previous section is in some cases cose to best possibe. Theorem 6 There is c > 0 such that for each sufficienty arge, there is a 3-uniform hypergraph H with maximum degree at most for which the 4-coour Ramsey number of H satisfies where n is the number of vertices of H. r(h, H, H, H) 2 2c n, Proof: Our proof uses the same 4-edge-coouring of the compete 3-uniform hypergraph that was constructed by Erdős and Hajna (see, e.g., [17]). Not ony does this coouring have no arge monochromatic compete 3-uniform hypergraph, but we show it aso does not have any monochromatic copies of a much sparser 3-uniform hypergraph H. Let n 4 be even, m = 2 n/4, and suppose the edges of the compete graph K m are cooured red or bue in such a way that neither coour contains a monochromatic copy of the graph K n/2. Such an edge-coouring exists by the ower bound of Erdős (see [17]) on the Ramsey number of the compete graph. Let V = {v 1,, v n } be a set of vertices and et H be the 3-uniform hypergraph on V whose edge set is given by {v i, v i+1, v j } for a 1 i, j n. (Note that when i = n, we consider i + 1 to be equa to 1.) It is straightforward to check that every vertex in H has degree 3n. We are going to define a 4-coouring of the compete 3-uniform hypergraph on the set T = {(γ 1,, γ m ) : γ i = 0 or 1} in such a way that there is no monochromatic copy of H. Note that then we wi be done, since T has size 2 m 2 2n/4 whie H has maximum degree 3n. To define our coouring, we need some definitions: 9

10 If ɛ = (γ 1,, γ m ), ɛ = (γ 1,, γ m) and ɛ ɛ, define δ(ɛ, ɛ ) = max{i : γ i γ i}, that is, δ(ɛ, ɛ ) is the argest coordinate at which they differ. We can now define an ordering on T by ɛ < ɛ if γ i = 0, γ i = 1, ɛ < ɛ if γ i = 1, γ i = 0. Another way of ooking at this ordering is to assign to each ɛ the number b(ɛ) = m i=1 γ i2 i 1. The ordering then says simpy that ɛ < ɛ iff b(ɛ) < b(ɛ ). It is important to note the foowing two properties of the function δ: (a) if ɛ 1 < ɛ 2 < ɛ 3, then δ(ɛ 1, ɛ 2 ) δ(ɛ 2, ɛ 3 ); (b) if ɛ 1 < ɛ 2 < < ɛ r, then δ(ɛ 1, ɛ r ) = max 1 i r 1 δ(ɛ i, ɛ i+1 ). Now we are ready to define our coouring of the compete 3-uniform hypergraph τ on vertex set T. To begin, suppose that {ɛ 1, ɛ 2, ɛ 3 } with ɛ 1 < ɛ 2 < ɛ 3 is an edge in τ. Write δ 1 = δ(ɛ 1, ɛ 2 ), δ 2 = δ(ɛ 2, ɛ 3 ). Then we coour as foows: C 1, if {δ 1, δ 2 } is red and δ 1 < δ 2 ; C 2, if {δ 1, δ 2 } is red and δ 1 > δ 2 ; C 3, if {δ 1, δ 2 } is bue and δ 1 < δ 2 ; C 4, if {δ 1, δ 2 } is bue and δ 1 > δ 2. Now, et S = {ɛ 1,, ɛ n } < be an ordered n-tupe within τ and suppose that there is a copy of H on S which is cooured by C 1. Suppose that the natura cyce {v 1,, v n } associated with H occurs as {ɛ π(1),, ɛ π(n) } where π is a permutation of 1,, n. For each i, 1 i n, et φ(i) = max(π(i), π(i + 1)) and ψ(i) = min(π(i), π(i + 1)). We caim that δ φ(i) 1 = δ(ɛ φ(i) 1, ɛ φ(i) ) must be arger than δ j = δ(ɛ j, ɛ j+1 ) for a j < φ(i) 1. First consider the tripe {ɛ ψ(i), ɛ φ(i) 1, ɛ φ(i) } <, which is an edge of the copy of H on S. The coouring C 1 impies that δ φ(i) 1 = δ(ɛ φ(i) 1, ɛ φ(i) ) > δ(ɛ ψ(i), ɛ φ(i) 1 ) = max δ j. ψ(i) j<φ(i) 1 This proves the caim for ψ(i) j < φ(i) 1. Next consider the tripe {ɛ j, ɛ ψ(i), ɛ φ(i) } < with j < ψ(i), which is aso an edge of the copy of H on S. The coouring C 1 impies that δ j δ(ɛ j, ɛ ψ(i) ) < δ(ɛ ψ(i), ɛ φ(i) ) = δ φ(i) 1. This proves the caim in the remaining cases 1 j < φ(i) 1. Consider the set {φ(2i 1)} n/2 i=1, which contains n/2 distinct eements since φ(i) = max(π(2i 1), π(2i)) and these pairs are disjoint. Let j 1,..., j n/2 be a permutation of the odd numbers up to n 1 such that φ(j 1 ) <... < φ(j n/2 ). By the caim in the previous paragraph, we have δ φ(j1 ) 1 < < δ φ(jn/2 ) 1. Consider, for each r < s with r, s {1,, n/2}, the tripe {ɛ ψ(jr), ɛ φ(jr), ɛ φ(js)} <, which is an edge of 10

11 the copy of H on S. Since ψ(j r ) < φ(j r ) < φ(j s ), by property (b) of function δ and the caim above, δ(ɛ ψ(jr), ɛ φ(jr)) = δ φ(jr) 1 and δ(ɛ φ(jr), ɛ φ(js)) = δ φ(js) 1. Therefore, by the definition of C 1 we must have that {δ φ(jr) 1, δ φ(js) 1} is red. Hence we get a cique of size n/2 in our origina coouring. But this cannot happen so we have a contradiction. A other cases foow simiary, so we re done. This resut is cosey reated to another interesting question: what is the maximum of r(h) over a k-uniform hypergraphs with m edges (we assume here that the hypergraphs we consider do not have isoated vertices)? For graphs, this question was posed by Erdős and Graham [10] who conjectured that the Ramsey number of a compete graph is at east the Ramsey number of every graph with the same number of edges. As noted by Erdős [9], this conjecture impies that there is a constant c such that for a graphs G, r(g) 2 c e(g). The best resut in this direction, proven by Aon, Kriveevich and Sudakov [1], is that r(g) 2 c e(g) og e(g). For hypergraphs, one can naturay ask a question simiar to the Erdős-Graham conjecture, i.e, is there a constant c = c(k) such that for every k-uniform hypergraph H, r(h) t k (c k e(h))? The proof of Theorem 6 has the foowing coroary: Coroary 2 There is a positive constant c such that for each positive integer m, there is a 3- uniform hypergraph H with at most m edges such that the 4-coour Ramsey number of H satisfies r(h, H, H, H) 2 2c m. Indeed the 3-uniform hypergraph H constructed in the proof of Theorem 6 has n vertices and ess than n 2 edges, whie r(h, H, H, H) t 3 (n/4) t 3 ( e(h)/4). This coroary demonstrates that the muticoour version of the hypergraph anaogue of the Erdős-Graham conjecture is fase. In the other direction, we prove the foowing theorem: Theorem 7 The q-coour Ramsey number of any k-uniform hypergraph H with m edges satisfies for k 4, and r(h,, H) t k (c m) r(h,, H) t 3 (c m og m) for k = 3, where constant c depends ony on k and q. Theorem 7 foows immediatey from the remark after the proof of Theorem 4, the fact that the maximum degree of H is ceary at most m, and the foowing emma. Lemma 2 Every k-uniform hypergraph H with m edges has strong chromatic number at most k m. Proof: Let H be the graph on the same vertex set as H with two vertices adjacent if they ie in an edge of H. The strong chromatic number of H is ceary equa to the chromatic number of H. The number e(h ) of edges of H is at most ( ) ( k 2 m k m ) ( 2 since each edge of H gives rise to at most k 2) edges ( of H. To finish the proof, note that the chromatic number χ of any graph with t edges satisfies χ ) 2 t because in an optima coouring there shoud be an edge between any two coour casses. 11

12 5 Concusion Throughout this paper we have aimed for simpicity in the exposition. Accordingy, in proving Theorem 2, we have cut some corners to make the proof as pithy as possibe. The resuting constant, c = (2k) k 1, is doubtess far from best possibe, but we beieve that this oss is outweighed by the resuting brevity of exposition. As we noted in the introduction, our Theorem 4 impies that, for k 4, there exists a constant c = c(k) such that, for any graph H on n vertices with maximum degree, r(h) t k (c )n, where the constant c depends ony on k. For k = 3, however, it ony impies that r(h) 2 2c og n, (1) which coud perhaps be improved a itte. It is worth noting aso that for k = 2, the best known bound, proved by Graham, Röd and Ruciński [15] using a very different method is r(h) 2 c og2 n. (2) In ight of the situation for higher k as we as the ower bound constructions for k = 2, 3, the foowing is a natura question: Probem 1 Can the og factors in the highest exponent of the upper bounds (1) and (2) be removed? This probem is certainy difficut in the k = 2 case, but maybe a different extension of the methods of [12] or an appropriate generaisation of the work of Graham, Röd and Ruciński coud resove the k = 3 case. It aso seems ikey to us that the ower bound for this probem is essentiay the same as the upper bound. So we have the foowing open probem: Probem 2 Is it true that for a k and and sufficienty arge n, there exists a k-uniform hypergraph H with maximum degree and n vertices such that r(h) t k (c )n, where c > 0 ony depends on k? References [1] N. Aon, M. Kriveevich, B. Sudakov: Turán numbers of bipartite graphs and reated Ramsey-type questions, Combin. Probab. Comput. 12 (2003), [2] S.A. Burr, P. Erdős: On the magnitude of generaized Ramsey numbers for graphs, in: Infinite and Finite Combinatorics, vo. 1, Cooq. Math. Soc. János Boyai 10 (1975) [3] V. Chvatá, V. Röd, E. Szemerédi, W.T. Trotter Jr.: The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), [4] D. Conon: A new upper bound for diagona Ramsey numbers, Annas of Mathematics, to appear. [5] D. Conon: Hypergraph Packing and sparse bipartite Ramsey numbers, submitted. [6] O. Cooey, N. Fountouakis, D. Kühn, D. Osthus: 3-uniform hypergraphs of bounded degree have inear Ramsey numbers, to appear in J. Combin. Theory Ser. B. 12

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