Given bipartite graphs $H_1$, \dots , $H_k$, the bipartite Ramsey number $br(H_1,\dots, H_k)$ is the minimum integer $N$ such that any $k$-edge-coloring of complete bipartite graph $K_{ N, N}$ contains a monochromatic $H_i$ in color $i$ for $1\le i\le k$. There are considerable results on asymptotic values of bipartite Ramsey numbers of cycles. For exact value, Zhang-Sun \cite{ Zhangs} determined $br(C_4, C_{ 2n})$, Zhang-Sun-Wu \cite{ Zhangsw} determined $br(C_6, C_{ 2n})$, and Gholami-Rowshan \cite{ GR} determined $br(C_8, C_{ 2n})$. In this paper, we solve all remaining cases and give the exact values of $br(C_{ 2n}, C_{ 2m})$ for all $n\ge m\ge 5$, this answers a question concerned by Buci\'c-Letzter-Sudakov \cite{ BLS}, Gholami-Rowshan \cite{ GR}, Zhang-Sun \cite{ Zhangs}, and Zhang-Sun-Wu \cite{ Zhangsw}.
Categories: math.CO