The $m$-bipartite Ramsey number of the $K_{2,2}$ versus $K_{6,6}$

Abstract

Given bipartite graphs $G_1, \ldots, G_n$, the bipartite Ramsey number $BR(G_1,\ldots, G_n)$ is the last integer $b$ such that any complete bipartite graph $K_{b,b}$ with edges coloured with colours $1,2,\ldots,n$ contains a copy of some $G_i$ ($1\leq i\leq n$) where all edges of $G_i$ have colour $i$. As another view of bipartite Ramsey numbers, for given bipartite graphs $G_1, \ldots, G_n$ and a positive integer $m$, the $m$-bipartite Ramsey number $BR_m(G_1, \ldots, G_n)$, is defined as the least integer $b$, such that any complete bipartite graph $K_{m,b}$ with edges coloured with colours $1,2,\ldots,n$ contains a copy of some $G_i$ ($1\leq i\leq n$) where all edges of $G_i$ have colour $i$. The size of $BR_m(G_1, G_2)$, where $G_1=K_{2,2}$ and $G_2\in \{K_{3,3}, K_{4,4}\}$ for each $m$ and the size of $BR_m(K_{3,3}, K_{3,3})$ and $BR_m(K_{2,2}, K_{5,5})$ for special values of $m$, have been determined in several article up to now. In this article, we compute the size of $BR_m(K_{2,2}, K_{6,6})$ for some $m\geq 2$.