The Bipartite Ramsey Numbers $b(C_{2m};K_{2,2})$

Abstract

Given bipartite graphs $H_1$ and $H_2$, the bipartite Ramsey number $b(H_1; H_2)$ is the smallest integer $b$ such that any subgraph $G$ of the complete bipartite graph $K_{b,b}$, either $G$ contains a copy of $H_1$ or its complement relative to $K_{b,b}$ contains a copy of $H_2$. It is known that $b(K_{2,2};K_{2,2})=5, b(K_{2,3};K_{2,3})=9, b(K_{2,4};K_{2,4})=14$ and $b(K_{3,3};K_{3,3})=17$. In this paper we study the case $H_1$ being even cycles and $H_2$ being $K_{2,2}$, prove that $b(C_6;K_{2,2})=5$ and $b(C_{2m};K_{2,2})=m+1$ for $m\geq 4$.