The characterization of zero-sum (mod 2) bipartite Ramsey numbers

Abstract

Let G be a bipartite graph, with k|e(G). The zero-sum bipartite Ramsey number B(G, Zk) is the smallest integer t such that in every Zk-coloring of the edges of Kt,t, there is a zero-sum mod k copy of G in Kt,t. In this article we give the first proof that determines B(G, Z2) for all possible bipartite graphs G. In fact, we prove a much more general result from which B(G, Z2) can be deduced: Let G be a (not necessarily connected) bipartite graph, which can be embedded in Kn,n, and let F be a field. A function f : E(Kn,n) → F is called G-stable if every copy of G in Kn,n has the same weight (the weight of a copy is the sum of the values of f on its edges). The set of all G-stable functions, denoted by U(G, Kn,n, F) is a linear space, which is called the Kn,n uniformity space of G over F. We determine U(G, Kn,n, F) and its dimension, for all G, n and F. Utilizing this result in the case F = Z2, we can compute B(G, Z2), for all bipartite graphs G. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 151–166, 1998