2018/08/30 by Shaoqiang Liu, Liu, Shaoqiang, Yuejian Peng +1
Mathematics · #05C35 #05C38 #Advanced Topology and Set Theory #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1808.10127
openalex publication_date 2018/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For an integer r≥ 2 and bipartite graphs Hi, where 1≤ i≤ r, the bipartite Ramsey number br(H1,H2,…,Hr) is the minimum integer N such that any r-edge coloring of the complete bipartite graph KN,N contains a monochromatic subgraph isomorphic to Hi in color i for some i, 1≤ i≤ r. We show that for α1,α2>0, br(C2\lfloor α1 n\rfloor,C2\lfloor α2 n\rfloor)=(α1+α2+o(1))n. We also show that if r≥ 3, α1,α2>0, αj+2≥ [(j+2)!-1]∑j+1i=1 αi for j=1,2,…,r-2, then br(C2\lfloor α1 n\rfloor,C2\lfloor α2 n\rfloor,…,C2\lfloor αr n\rfloor)=(∑rj=1 αj+o(1))n. For ξ>0 and sufficiently large n, let G be a bipartite graph with bipartition \V1,V2\, |V1|=|V2|=N, where N=(2+8ξ)n. We prove that if δ(G)>((7)/(8)+9ξ)N, then any 2-edge coloring of G contains a monochromatic copy of C2n.