Bipartite Ramsey numbers of cycles

Abstract

Let H1, …, Hk be bipartite graphs. The bipartite Ramsey number br(H1,…,Hk) is the minimum integer N such that any k-edge-coloring of the complete bipartite graph KN,N contains a monochromatic Hi in color i for some i∈{1,2,…,k}. The study of bipartite Ramsey number was initiated in the early 70s by Faudree and Schelp (1975), and they showed that br(P2n,P2m)=n+m−1, where Pn denotes a path with n vertices. Combining their result and the regularity lemma, one can obtain asymptotic value br(C2⌊α1n⌋,C2⌊α2n⌋)=(α1+α2+o(1))n for α1,α2>0, where Cn denotes a cycle with n vertices. The study of bipartite Ramsey numbers of cycles has also attracted lots of attention recently. For example, Bucić, Letzter and Sudakov (2019) showed that br(C2n,C2n,C2n)=(3+o(1))n and they remarked that it is interesting to determine the exact value of br(C2n,C2m). Zhang and Sun (2011), Zhang, Sun, and Wu (2013), and Gholami and Rowshan (2021) determined br(C4,C2n), br(C6,C2n), and br(C8,C2n) respectively. In this paper, we determine br(C2n,C2m) for all the remaining cases and show that br(C2n,C2m)=n+m−1 for all n>m≥5 and br(C2n,C2n)=2n if n≥5. It is somehow interesting that br(C2n,C2m) is the same as br(P2n,P2m) if n≠m, and br(C2n,C2n) is one more than br(P2n,P2n).