The m-Bipartite Ramsey Number of the K{2,2} Versus K{6,6}

For the given bipartite graphs G1, . . . , Gn, the bipartite Ramsey number BR(G1, . . . , Gn) is the least positive integer b such that any complete bipartite graph K b,b having edges coloured with 1, 2, . . . , n, contains a copy of some Gi (1 i n), where all the edges of Gi have colour i. For the given bipartite graphs G1, . . . , Gn and a positive integer m, the m-bipartite Ramsey number BRm(G1, . . . , Gn) is defined as the least positive integer b (b m) such that any complete bipartite graph K m,b having edges coloured with 1, 2, . . . , n, contains a copy of some Gi (1 i n), where all the edges of Gi have colour i. The values of BRm(G1, G2) (for each m), BRm(K3,3, K3,3) and BRm(K2,2, K5,5) (for particular values of m) have already been determined in several articles, where G1 = K2,2 and G2 {K3,3, K4,4}. In this article, the value of BRm(K2,2, K6,6) is computed for each m {2, 3, . . . , 8}.