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Improved Bounds for the Ramsey Number of Tight Cycles Versus Cliques

Part of: Graph theory Extremal combinatorics

Published online by Cambridge University Press:  08 March 2016

DHRUV MUBAYI
DHRUV MUBAYI*
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607, USA (e-mail: mubayi@uic.edu)
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Abstract

The 3-uniform tight cycle Cs3 has vertex set ${\mathbb Z}_s$ and edge set {{i, i + 1, i + 2}: i ∈ ${\mathbb Z}_s$}. We prove that for every s ≢ 0 (mod 3) with s ⩾ 16 or s ∈ {8, 11, 14} there is a cs > 0 such that the 3-uniform hypergraph Ramsey number r(Cs3, Kn3) satisfies

$$\begin{equation*} r(C_s^3, K_n^3)< 2^{c_s n \log n}.\ \end{equation*}$$
This answers in a strong form a question of the author and Rödl, who asked for an upper bound of the form $2^{n^{1+\epsilon_s}}$ for each fixed s ⩾ 4, where εs → 0 as s → ∞ and n is sufficiently large. The result is nearly tight as the lower bound is known to be exponential in n.

MSC classification

Primary: 05C65: Hypergraphs
Secondary: 05D10: Ramsey theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 5 , September 2016 , pp. 791 - 796
Copyright
Copyright © Cambridge University Press 2016 

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References

[1] Ajtai, M., Komlós, J. and Szemerédi, E. (1980) A note on Ramsey numbers. J. Combin. Theory Ser. A 29 354–360.CrossRefGoogle Scholar
[2] Bohman, T. and Keevash, P. (2010) The early evolution of the H-free process. Invent. Math. 181 291–336.Google Scholar
[3] Bohman, T. and Keevash, P. (2013) Dynamic concentration of the triangle-free process. The Seventh European Conference on Combinatorics, Graph Theory and Applications, (Norm, P.. ed.), vol. 16, CRM Series, pp. 489–495.CrossRefGoogle Scholar
[4] Caro, Y., Li, Y., Rousseau, C. and Zhang, Y. (2000) Asymptotic bounds for some bipartite graph: Complete graph Ramsey numbers. Discrete Math. 220 51–56.Google Scholar
[5] Conlon, D., Fox, J. and Sudakov, B. (2010) Hypergraph Ramsey numbers. J. Amer. Math. Soc. 23 247–266.CrossRefGoogle Scholar
[6] Erdős, P. (1984) Extremal problems in number theory, combinatorics and geometry. In Proc. International Congress of Mathematicians, PWN, Warsaw, pp. 51–70.Google Scholar
[7] Erdős, P. and Hajnal, A. (1972) On Ramsey like theorems, problems and results. In Combinatorics: Proc. Conference Combinatorial Mathematics, IMA, Southend-on-Sea, pp. 123–140.Google Scholar
[8] Fiz Pontiveros, G., Griffiths, S. and Morris, R. The triangle-free process and R(3,k). arXiv:1302.6279 Google Scholar
[9] Kim, J. H. (1995) The Ramsey number R(3,t) has order of magnitude t 2/log t. Random Struct. Alg. 7 173–207.Google Scholar
[10] Kostochka, A., Mubayi, D. and Verstraëte, J. (2014) On independent sets in hypergraphs. Random Struct. Alg. 44 224–239.CrossRefGoogle Scholar
[11] Mubayi, D. and Rödl, V. (2016) Hypergraph Ramsey numbers: Tight cycles versus cliques. Bull. Lond. Math. Soc. 48 127–134.Google Scholar
[12] Spencer, J. (1972) Turán theorem for k-graphs. Discrete Math. 2 183–186.Google Scholar
[13] Sudakov, B. (2005) A new lower bound for a Ramsey-type problem. Combinatorica 25 487–498.CrossRefGoogle Scholar