The $m$-bipartite Ramsey number $B{R}_m(K_{2,2},K_{5,5})$

Abstract

The bipartite Ramsey number $BR(H_1,H_2,\dots ,H_k)$, is the smallest positive integer $b$, such that each $k$-decomposition of $E(K_{b,b})$ contains $H_i$ in the $i$-th class for some $i,1\le i\le k$. As another view of bipartite Ramsey numbers, for given two bipartite graphs $H_1$ and $H_2$ and a positive integer $m$, the $m$-bipartite Ramsey number $B{R}_m(H_1,H_2)$, is defined as the least integer $n$, such that any subgraph of $K_{m,n}$ say $H$, results in $H_1\subseteq H$ or $H_2\subseteq \overline{H}$. The size of $B{R}_m(K_{2,2},K_{3,3})$, $B{R}_m(K_{2,2},K_{4,4})$ for each $m$, and the size of $B{R}_m(K_{3,3},K_{3,3})$ for some $m$, have been determined in several papers up to now. Also, it is shown that $BR(K_{2,2},K_{5,5})=17$. In this article, we compute the size of $B{R}_m(K_{2,2},K_{5,5})$ for some $m\ge 2$.