The m-Bipartite Ramsey Number BRm(H1, H2)

Abstract In a (G1, G2) coloring of a graph G, every edge of G is in G1 or G2. For two bipartite graphs H1 and H2, the bipartite Ramsey number BR(H1, H2) is the least integer b ≥ 1, such that for every (G1, G2) coloring of the complete bipartite graph Kb,b, results in either H1 ⊆ G1 or H2 ⊆ G2. As another view, for bipartite graphs H1 and H2 and a positive integer m, the m-bipartite Ramsey number BRm(H1, H2) of H1 and H2 is the least integer n (n ≥ m) such that every subgraph G of Km,n results in H1 ⊆ G or H2 ⊆ Ḡ. The size of m-bipartite Ramsey number BRm(K2,2, K2,2), the size of m-bipartite Ramsey number BRm(K2,2, K3,3) and the size of m-bipartite Ramsey number BRm(K3,3, K3,3) have been computed in several articles up to now. In this paper we determine the exact value of BRm(K2,2, K4,4) for each m ≥ 2.