On the size-Ramsey number of cycles

Javadi, R. and Khoeini, F. and Omidi, G.R. and Pokrovskiy, Alexey (2019) On the size-Ramsey number of cycles. Combinatorics, Probability, and Computing 28 (6), pp. 871-880. ISSN 0963-5483.

Abstract

For given graphs $G_1,\ldots,G_k$, the size-Ramsey number $\hat{R}(G_1,\ldots,G_k)$ is the smallest integer $m$ for which there exists a graph $H$ on $m$ edges such that in every $k$-edge coloring of $H$ with colors $1,\ldots,k$, $ H $ contains a monochromatic copy of $G_i$ of color $i$ for some $1\leq i\leq k$. We denote $\hat{R}(G_1,\ldots,G_k)$ by $\hat{R}_{k}(G)$ when $G_1=\cdots=G_k=G$. Haxell, Kohayakawa and \L{}uczak showed that the size-Ramsey number of a cycle $C_n$ is linear in $n$ i.e. $\hat{R}_{k}(C_{n})\leq c_k n$ for some constant $c_k$. Their proof, however, is based on the regularity lemma of Szemer\'{e}di and so no specific constant $c_k$ is known. In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of $\hat{R}_{k}(C_{n})\leq c_k n$, avoiding the use of the regularity lemma, where $ c_k $ is exponential and doubly-exponential in $ k $, when $ n $ is even and odd, respectively. In particular, we show that for sufficiently large $n$ we have $\hat{R}(C_{n},C_{n}) \leq 10^5\times cn,$ where $c=6.5$ if $n$ is even and $c=1989$ otherwise.

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