Random bipartite Ramsey numbers of long cycles

Abstract

For graphs G and H, let G⟶kH signify that any k-edge coloring of G contains a monochromatic H as a subgraph. Let G(K2(N),p) be random graph spaces with edge probability p, where K2(N) is the complete N×N bipartite graph. Let C2n be a cycle of length 2n and let k=2,3. It is shown for any ϵ>0, there exists T=T(ϵ)>0 such that if np>T, then Pr[G(K2((k+ϵ)n),p)⟶kC2n]→1 as n→∞, for which the proof relies heavily on the sparse regularity lemma for multipartite graphs.