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

1998, Journal of Graph Theory

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-coloring of the edges of K t,t , there is a zero-sum mod k copy of G in K t,t. In this paper we give the first proof which determines B(G, Z 2) for all possible bipartite graphs G. In fact, we prove a much more general result from which B(G, Z 2) can be deduced: Let G be a (not necessarily connected) bipartite graph, which can be embedded in K n,n , and let F be a field. A function f : E(K n,n) → F is called G-stable if every copy of G in K n,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, K n,n , F) is a linear space which is called the K n,n uniformity space of G over F. We determine U (G, K n,n , F) and its dimension, for all G, n and F. Utilizing this result in the case F = Z 2 , we can compute B(G, Z 2), for all bipartite graphs G. 1 Introduction All graphs and hypergraphs considered here are finite, undirected and have no loops or multiple edges. For the standard graph-theoretic notations the reader is referred to [5]. Let Z k denote the cyclic additive group of order k. A Z k-coloring of the edges of a graph G = (V, E) is a function f : E(G) → Z k. If e∈E(G) f (e) = 0 in Z k , we say that G is a zero-sum graph mod k with respect to f. The concepts of zero-sum Ramsey numbers and bipartite zero-sum Ramsey numbers were first introduced by Bialostocki and Dierker in [3] and [2], and by Caro in [7]. If k | e(G) then the zero-sum Ramsey number R(G, Z k) is the smallest integer t such that in every Z k-coloring of K t there exists a zero-sum mod k copy of G in K t. If k | e(G) and G is bipartite then the zero-sum bipartite Ramsey number B(G, Z k) is the smallest integer t such that in every