The bipartite Ramsey number br(C_2n, C_2m)

2021/12/30 by Zilong Yan, Yan, Zilong, Yuejian Peng +1 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2112.14960

openalex publication_date 2021/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given bipartite graphs H₁, … , H_k, the bipartite Ramsey number br(H₁,…, 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≤ i≤ k. There are considerable results on asymptotic values of bipartite Ramsey numbers of cycles. For exact value, Zhang-Sun \citeZhangs determined br(C₄, C_2n), Zhang-Sun-Wu \citeZhangsw determined br(C₆, C_2n), and Gholami-Rowshan \citeGR determined br(C₈, C_2n). In this paper, we solve all remaining cases and give the exact values of br(C_2n, C_2m) for all n≥ m≥ 5, this answers a question concerned by Bucić-Letzter-Sudakov \citeBLS, Gholami-Rowshan \citeGR, Zhang-Sun \citeZhangs, and Zhang-Sun-Wu \citeZhangsw.

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