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Improved Bounds for the Ramsey Number of Tight Cycles Versus Cliques

Part of: Extremal combinatorics Graph theory

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 Cs 3 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(Cs 3, Kn 3) 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

Information

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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