Ramsey theorems for multiple copies of graphs

If G and H are graphs, define the Ramsey number r(G, H) to be the least number p such that if the edges of the complete graph K_pare colored red and blue (say), either the red graph contains G as a subgraph or the blue graph contains H. Let mG denote the union of m disjoint copies of G. The following result is proved: Let G and H have k and l points respectively and have point independence numbers of i and j respectively. Then N - 1 ≤ r(mG, nH) ≤ N + C, where N = km + In - min (mi, mj) and where C is an effectively computable function of G and H. The method used permits exact evaluation of r(mGt nH) for various choices of G and H, especially when m — n or G-H. In particular, r(mK₃, nK₃) = 3m + 2n when m ≥ n, m ≥ 2.