BIPARTITE RAMSEY NUMBER PAIRS INVOLVING CYCLES

For bipartite graphs G(1), G(2), ... , G(k), the bipartite Ramsey number b(G(1), G(2), ... , G(k)) is the least positive integer b, so that any coloring of the edges of K(b,b )with k colors, will result in a copy of G(i) 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 K-r,K-t has a monochromatic copy of C-2s. Let a and b be positive integers with a >= b. For bipartite graphs G(1) and G(2), the bipartite Ramsey number pair (a, b), denoted by b(p)(G(1), G(2)) = (a, b), is an ordered pair of integers such that for any blue-red coloring of the edges of K-r,K-t, with r >= t, either a blue copy of G(1) exists or a red copy of G(2) 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 b(p)(P-2s, P-2s) = (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 K-2s,(2s-1) has a monochromatic C-2s. This will imply that b(p)(C-2s, C-2s) = (2s, 2s - 1), if s is sufficiently large.