THE CHARACTERIZATION OF ZERO-SUM (MOD-2) BIPARTITE RAMSEY NUMBERS

Let G be a bipartite graph, with k\e(G). The zero-sum bipartite Ramsey number B(G, Z(k)) is the smallest integer t such that in every Z(k)-c oloring of the edges of K-t,K-t, there is a zero-sum mod k copy of G i n K-t,K-t. In this article we give the first proof that determines B(G , Zz) for all possible bipartite graphs G. In fact, we prove a much mo re general result from which B(G, Zz) can be deduced: Let G be a (not necessarily connected) bipartite graph, which can be embedded in K-n,K -n, and let F be a field. A function f : E(K-n,K-n) --> F is called G- stable if every copy of G in K-n,K-n has the same weight (the weight o f a copy is the sum of the values of f on its edges). The set of all G -stable functions, denoted by U(G, K-n,K-n, F) is a linear space, whic h is called the K-n,K-n uniformity space of G over F. We determine U(G , K-n,K-n, F) and its dimension, for all G, n and F. Utilizing this re sult in the case F = Z(2), we can compute B(G, Z(2)), for all bipartit e graphs G. (C) 1998 John Wiley & Sons, Inc.