Random bipartite Ramsey numbers of long cycles

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