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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 bipartitegraph $K_{b,b}$ with edges coloured with colours $1,2,\ldots,n$ contains a copyof some $G_i$ ($1\leq i\leq n$) where all edges of $G_i$ have colour $i$. Asanother 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 anycomplete 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 edgesof $G_i$ 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 beendetermined in several article up to now. In this article, we compute the sizeof $BR_m(K_{2,2}, K_{6,6})$ for some $m\geq 2$.

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