Ramsey regions

Let (T1, T2, . . . , Tc) be a fixed c-tuple of sets of graphs (i.e. each Ti is a set of graphs). Let R(c, n, (T1, T2, . . . , Tc)) denote the set of all n-tuples, (a1, a2, . . . , an), such that every c-coloring of the edges of the complete multipartite graph, Ka1,a2,...,an , forces a monochromatic subgraph of color i from the set Ti (for at least one i). If N denotes the set of non-negative integers, then R(c, n, (T1, T2, . . . , Tc)) ⊆ Nn. We call such a subset of Nn a “Ramsey region”. An application of Ramsey’s Theorem shows that R(c, n, (T1, T2, . . . , Tc)) is non-empty for n?0. For a given c-tuple, (T1, T2, . . . , Tc), known results in Ramsey theory help identify values of n for which the associated Ramsey regions are non-empty and help establish specific points that are in such Ramsey regions. In this paper, we develop the basic theory and some of the underlying algebraic structure governing these regions. © 2007 Elsevier B.V. All rights reserved.