On the size-Ramsey number of cycles
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 Łuczak 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, is based on the regularity lemma of Szemerédi 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 give an alternative proof of $\hat{R}_{k}(C_{n})\leq c_k n$, avoiding the use of the regularity lemma. For two colours, we show that for sufficiently large $n$ we have $\hat{R}(C_{n},C_{n}) \leq 10^6\times cn,$ where $c=843$ if $n$ is even and $c=113482$ otherwise.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2017
- DOI:
- 10.48550/arXiv.1701.07348
- arXiv:
- arXiv:1701.07348
- Bibcode:
- 2017arXiv170107348J
- Keywords:
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- Mathematics - Combinatorics;
- 05C55;
- 05D10