The bipartite Ramsey number $br(C_{2n},C_{2m})$

Abstract

Given bipartite graphs $H_1$, \dots , $H_k$, the bipartite Ramsey number $br(H_1,\dots ,H_k)$ is the minimum integer $N$ such that any $k$-edge-coloring of complete bipartite graph $K_{N,N}$ contains a monochromatic $H_i$ in color $i$ for $1\le i\le k$. There are considerable results on asymptotic values of bipartite Ramsey numbers of cycles. For exact value, Zhang-Sun \cite{Zhangs} determined $br(C_4,C_{2n})$, Zhang-Sun-Wu \cite{Zhangsw} determined $br(C_6,C_{2n})$, and Gholami-Rowshan \cite{GR} determined $br(C_8,C_{2n})$. In this paper, we solve all remaining cases and give the exact values of $br(C_{2n},C_{2m})$ for all $n\ge m\ge 5$, this answers a question concerned by Bucić-Letzter-Sudakov \cite{BLS}, Gholami-Rowshan \cite{GR}, Zhang-Sun \cite{Zhangs}, and Zhang-Sun-Wu \cite{Zhangsw}.