The Multicolour Size--Ramsey Number of an Even Cycle

Abstract

We determine the $k$-colour size--Ramsey number of even cycles up to absolute constant factors. For every $k\ge 2$ and every even $n\ge 100\log k$, \[ \widehat R_k(C_n)=Θ(k^2\log k)n. \] The lower bound follows from the corresponding result of Beke, Li and Sahasrabudhe for paths, while our upper bound improves the previous best estimate $O(k^{34}n)$ of Javadi, Kohayakawa and Miralaei.