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 in...