On the sizeRamsey number of cycles
Javadi, R. and Khoeini, F. and Omidi, G.R. and Pokrovskiy, Alexey (2019) On the sizeRamsey number of cycles. Combinatorics, Probability, and Computing 28 (6), pp. 871880. ISSN 09635483.

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Abstract
For given graphs $G_1,\ldots,G_k$, the sizeRamsey 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 sizeRamsey 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 sizeRamsey 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 doublyexponential 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.
Metadata
Item Type:  Article 

School:  School of Business, Economics & Informatics > Economics, Mathematics and Statistics 
Depositing User:  Alexey Pokrovskiy 
Date Deposited:  07 Nov 2018 11:38 
Last Modified:  10 Jun 2021 17:48 
URI:  https://eprints.bbk.ac.uk/id/eprint/24980 
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