Another view of bipartite Ramsey numbers

For bipartite graphs F and H and a positive integer s, the s-bipartite Ramsey number BR s (F, H) of F and H is the smallest integer t with t s such that every red-blue coloring of K s, t results in a red F or a blue H. We evaluate this number for all positive integers s when F = K 2, 2 and H K 2, 3, K 3, 3.