BIPARTITE RAMSEY NUMBER PAIRS INVOLVING CYCLES

For bipartite graphs G1,G2, . . . ,Gk, the bipartite Ramsey number b(G1, G2, . . . ,Gk) is the least positive integer b, so that any coloring of the edges of Kb,b with k colors, will result in a copy of Gi in the ith color, for some i. We determine all pairs of positive integers r and t, such that for a sufficiently large positive integer s, any 2-coloring of Kr,t has a monochromatic copy of C2s. Let a and b be positive integers with a ≥ b. For bipartite graphs G1 and G2, the bipartite Ramsey number pair (a, b), denoted by bp(G1,G2) = (a, b), is an ordered pair of integers such that for any blue-red coloring of the edges of Kr,t, with r ≥ t, either a blue copy of G1 exists or a red copy of G2 exists if and only if r ≥ a and t ≥ b. In [Path-path Ramsey-type numbers for the complete bipartite graph, J. Combin. Theory Ser. B 19 (1975) 161-173], Faudree and Schelp showed that bp(P2s, P2s) = (2s - 1, 2s - 1), for s ≥ 1. In this paper we will show that for a sufficiently large positive integer s, any 2-coloring of K2s,2s-1 has a monochromatic C2s. This will imply that bp(C2s,C2s) = (2s, 2s - 1), if s is sufficiently large.