Bipartite Ramsey number pairs involving cycles

For bipartite graphs $G_1, G_2,\ldots,G_k$, the bipartite Ramsey number $b(G_1,$ $G_2,\ldots, G_k)$ is the least positive integer $b,$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors, will result in a copy of $G_i$ in the $i$th color, for some $i$. We determine all pairs of positive integers $r$ and $t$, such that for a sufficiently large positive integer $s$, any $2$-coloring of $K_{r,t}$ has a monochromatic copy of $C_{2s}$. Let $a$ and $b$ be positive integers with $a\geq b$. For bipartite graphs $G_1$ and $G_2$, the bipartite Ramsey number pair $(a,b)$, denoted by $b_p(G_1,G_2)=(a,b)$, is an ordered pair of integers such that for any blue-red coloring of the edges of $K_{r,t}$, with $r\geq t$, either a blue copy of $G_1$ exists or a red copy of $G_2$ exists if and only if $r\geq a$ and $t\geq b$. In [Path-path Ramsey-type numbers for the complete bipartite graph, J. Combin. Theory Ser. B 19 (1975) 161–173], Faudree and Schelp showed that $b_{p}(P_{2s},P_{2s})=(2s-1,2s-1)$, for $s\geq 1$. In this paper we will show that for a sufficiently large positive integer $s$, any 2-coloring of $K_{2s,2s-1}$ has a monochromatic $C_{2s}$. This will imply that $b_p(C_{2s}, C_{2s})=(2s,2s-1)$, if $s$ is sufficiently large.