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

TL;DR

This paper determines the exact bipartite Ramsey numbers for cycles of even length, completing the classification for all cases where both cycle lengths are at least 10.

Contribution

It provides the exact values of bipartite Ramsey numbers for pairs of even cycles, resolving all remaining open cases for cycles with length at least 10.

Findings

Exact bipartite Ramsey numbers for all even cycles with length ≥ 10.

Resolved previously open cases in bipartite Ramsey theory.

Answers a longstanding question in the field.

Abstract

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}.