The multicolor size-Ramsey numbers of cycles

Abstract

Given a positive integer r, the r-color size-Ramsey number of a graph H, denoted by Rˆ(H,r), is the smallest number of edges in a graph G such that in any edge coloring of G with r colors, G contains a monochromatic copy of H. Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle Cn is linear in n i.e. Rˆ(Cn,r)≤f(r)n, for some function f(r). Their proof, however, is based on the Szemerédi's regularity lemma and no explicit function f(r) is given there. Javadi, Khoeini, Omidi and Pokrovskiy gave an alternative proof for this result which avoids using the regularity lemma. Indeed, they proved that f(r) can be taken to be exponential and doubly exponential in r, in case n is even and odd, respectively.In this paper, we improve the upper bound f(r) to a polynomial function in r when n is even and to an exponential function in r when n is odd. We also prove that in the latter case, it cannot be improved to a polynomial bound in r. More precisely, we prove that there are some positive constants c,c′ such that for every even integer n, we have cr2n≤Rˆ(Cn,r)≤c′r120(log2⁡r)n and for every odd integer n, we have c2rn≤Rˆ(Cn,r)≤c′r2216r2n.