Vertex Ramsey Properties of Families of Graphs

For graphs F, G1, …, Gr, we write F→(G1, …, Gr) if for every coloring of the vertices of F with r colors there exists i, i=1, 2, …, r, such that a copy of Gi is colored with the ith color. For two families of graphs G1, …, Gr and H1, …, Hs, by (G1, …, Gr)→(H1, …, Hs) we denote the fact that F→(G1, …, Gr) implies F→(H1, …, Hs) for every graph F. In this paper, we give necessary and sufficient conditions for (G1, …, Gr)→(H1, …, Hs) under some weak assumptions on the families of G1, …, Gr and H1, …, Hs. We also consider the induced version of this problem.