The Ramsey Number for 3-Uniform Tight Hypergraph Cycles Article

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Combinatorics, Probability and Computing (2009) 18, 165–203. c○ 2009 Cambridge University Press doi:10.1017/S0963548309009675 Printed in the United Kingdom The Ramsey Number for 3-Uniform Tight Hypergraph Cycles P. E. HAXELL1∗ ,T. ̷LUCZAK2 ,Y.PENG3 ,V.RÖDL4† , A. R U C I ŃSKI2‡ and J. SKOKAN5§ 1Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 (e-mail: pehaxell@math.uwaterloo.ca) 2 Department of Discrete Mathematics, Adam Mickiewicz University, 61-614 Poznań, Poland (e-mail: tomasz@amu.edu.pl, andrzej@mathcs.emory.edu) 3Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN 47809, USA (e-mail: mapeng@isugw.indstate.edu) 4Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30032, USA (e-mail: rodl@mathcs.emory.edu) 5Department of Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK (e-mail: jozef@member.ams.org) Received 23 February 2007; revised 2 December 2008; first published online 4 February 2009 Let C (3) n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1,...,vn and edges v1v2v3, v2v3v4,...,vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C (3) n is asymptotically equal to 4n/3 ifn is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl. 1. Introduction Given a k-uniform hypergraph H, k � 2, the Ramsey number r(H) is the smallest integer N such that every red–blue colouring of the edges of the complete k-uniform hypergraph ∗ Partially supported by NSERC. † Supported by NSF grants DMS-0300529 and DMS-0800070. ‡ Corresponding author. Supported by the Polish Ministry grant N201036 32/2546. § Supported by NSF grant INT-0305793, NSA grant H98230-04-1-0035, and by FAPESP/CNPq grants (Proj. Temático–ProNEx Proc. FAPESP 2003/09925–5 and Proc. FAPESP 2004/15397-4).
166 P. E. Haxell, T. ̷Luczak, Y. Peng, V. Rödl, A. Ruciński and J. Skokan K (k) N with N vertices yields a monochromatic copy of H. A classical result in graph Ramsey theory ([1, 5, 18]) states that for k =2andn � 5 the Ramsey number of the graph cycle Cn with n vertices is r(Cn) = � 3 2n − 1 if n is even, 2n − 1 if n is odd. Thus, the Ramsey numbers for graph cycles depend strongly on the parity of n. In this paper we continue our study of Ramsey numbers for 3-uniform hypercycles, initiated in [11]. There are various definitions of a cycle in a 3-uniform hypergraph. Given a suitably labelled set of vertices {v1,...,vn}, aloose cycle has edge set {v1v2v3,v3v4v5,v5v6v7,...,vn−1vnv1}, while the tight cycle, denoted henceforth by C (3) n ,has the edge set {v1v2v3,v2v3v4,v3v4v5,...,vn−1vnv1,vnv1v2}. In [11] we proved that the Ramsey number for the n-vertex loose cycle, n even, is asymptotic to 5n/4. (Note that loose cycles do not exist for n odd.) Subsequently, Gyárfás, Sárközy and Szemerédi [8] extended this result to k-uniform loose cycles. Here an analogous problem is investigated for tight cycles. So far, the only known value of the Ramsey number for a tight cycle is r(C (3) 4 ) = 13 (see [16]). Asymptotically, it turns out that the Ramsey number for the tight cycle is larger than that for the loose cycle, and depends on whether n is divisible by 3. Thus, in this respect, tight cycles behave more like graph cycles than loose cycles do. Our aim is to prove the following theorem. Theorem 1.1. (a) For every integer n � 1 and i =0, 1, 2, r(C (3) � 4n − 1 if i =0, 3n+i ) � 6n +2i− 1 if i �= 0. (b) Let η>0 be given. Then for all sufficiently large n and i =0, 1, 2, r(C (3) � (4 + η)n if i =0, 3n+i ) � (6 + η)n if i �= 0. We should mention that one more natural definition of cycle for hypergraphs is the so-called Berge cycle. The Ramsey number for Berge cycles was investigated by Gyárfás, Sárközy and Szemerédi [9, 10]. The proof of part (a) and Theorem 1.1(b) also yield the asymptotic value of the Ramsey number of tight paths. A (tight) path P (3) n is a hypergraph with vertices v1,...,vn and edges v1v2v3, v2v3v4,...,vn−2vn−1vn. Corollary 1.2. r(P (3) n )=(4/3+o(1))n, whereo(1) → 0 as n →∞. The loose and tight cycles are examples of hypergraphs with bounded maximum degree. For this class of hypergraphs, it was conjectured that their Ramsey number is linear in
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