The bipartite Ramsey number $br(C_{2n}, C_{2m})$

Given bipartite graphs H1, . . . , Hk, the bipartite Ramsey number br(H1, . . . , Hk) is the minimum integer N such that any k-edge-coloring of complete bipartite graph KN,N contains a monochromatic Hi in color i for 1 ≤ i ≤ k. There are considerable results on asymptotic values of bipartite Ramsey numbers of cycles. For exact value, Zhang-Sun [28] determined br(C4, C2n), Zhang-Sun-Wu [29] determined br(C6, C2n), and Gholami-Rowshan [12] determined br(C8, C2n). In this paper, we solve all remaining cases and give the exact values of br(C2n, C2m) for all n ≥ m ≥ 5, this answers a question concerned by Bucić-Letzter-Sudakov [5], Gholami-Rowshan [12], Zhang-Sun [28], and Zhang-Sun-Wu [29].

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