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 Kp are 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 I points respectively and have point independence numbers of i and j respectively. Then N 1 5 r(mG, nH) < N + C, where N = km + In min(mi, ml) and where C is an effectively computable function of G and H. The method used permits exact evaluation of r(mG, nH) for various choices of G and H, especially when m = n or G = H. In particular, r(mK3 , nK 3 ) = 3m + 2n when m _> n, m 3 2 . 1 . Introduction . Let G and H be graphs without isolated points . Following Chvátal and Harary [11, define the Ramsey number r(G, H) to be the least integer n such that if the edges of Kn (the complete graph on n points) are two-colored, say red and blue, either the red graph contains G as a subgraph or the blue graph containsH. Note that r(Kk, KI) is the "ordinary" Ramsey number r(k, 1) for which an extensive literature exists . The evaluation of r(G, H) has received attention from several authors in recent years. An extensive survey is given in [2] . In this paper we will generally follow the notation of Harary [3] . In particular, let nG denote the union of n vertex-disjoint copies of G . In §2 we obtain surprisingly sharp and general upper and lower bounds for r(nG, nH) for G and H fixed and n sufficiently large. In §3 we extend these results to r(mG, nH), in §4 to k-graphs . In §5 we consider a related problem of J . W. Moon concerning the decomposition of a complete graph into complete monochromatic subgraphs of prescribed size . Finally, in §6 we give exact values for various cases . 2. The Ramsey numbers r(nG, IIH). Again following [3], let p(G) denote the number of points of G and let 9,(G) denote the number of points in a maximal independent set in G . As a special notation, let [X] 2 denote the complete graph on X and XY denote the complete bipartite graph on X and Y. Also, let r(G) = r(G, G) ; these we call the diagonal Ramsey numbers . Presented to the Society, January 16, 1974 ; received by the editors January 14, 1974 . AMS (MOS) subject classifications (1970) . Primary 05C35 ; Secondary 05C15 . ( 1 ) Supported under U .S. Office of Naval Research Contract N00014-67-A-0202-0063 . Copyright © 1975 . American Mathematical Society