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The bipartite Ramsey number $BR(H_1,H_2,\ldots,H_k)$, is the smallestpositive integer $b$, such that each $k$-decomposition of $E(K_{b,b})$ contains$H_i$ in the $i$-th class for some $i, 1\leq i\leq k$. As another view ofbipartite Ramsey numbers, for given two bipartite graphs $H_1$ and $H_2$ and apositive integer $m$, the $m$-bipartite Ramsey number $BR_m(H_1, H_2)$, isdefined 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$BR_m(K_{2,2}, K_{3,3})$, $BR_m(K_{2,2}, K_{4,4})$ for each $m$, and the sizeof $BR_m(K_{3,3}, K_{3,3})$ for some $m$, have been determined in severalpapers up to now. Also, it is shown that $BR(K_{2,2}, K_{5,5})=17$. In thisarticle, we compute the size of $BR_m(K_{2,2}, K_{5,5})$ for some $m\geq 2$.

The $m$-bipartite Ramsey number $BR_m(K_{2,2},K_{5,5})$