Multicolor Size-Ramsey Number of Cycles

Given a positive integer r, the r-color size-Ramsey number of a graph H , denoted by R̂(H, r), is the smallest integer m for which there exists a graph G with m edges such that, in any edge coloring of G with r colors, G contains a monochromatic copy of H . Haxell, Kohayakawa and Luczak showed that the size-Ramsey number of a cycle Cn is linear in n i.e. R̂(Cn, r) ≤ crn, for some constant cr. Their proof, however, is based on the Szemerédi’s regularity lemma and so no specific constant cr is known. Javadi, Khoeini, Omidi and Pokrovskiy gave an alternative proof for this result which avoids using of the regularity lemma. Indeed, they proved that if n is even, then cr is exponential in r and if n is odd, then cr is doubly exponential in r. In this paper, we improve the bound cr and prove that cr is polynomial in r when n is even and is exponential 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 c1, c2 such that for every even integer n, we have c1r n ≤ R̂(Cn, r) ≤ c2r(log r)n and for every odd integer n, we have c12 rn ≤ R̂(Cn, r) ≤ c22 +2 log rn.