Extremal Ramsey theory for graphs

If G and H are graphs, define the Ramsey number to be the least number p such that if the lines of the complete graph Kp are colored red and blue (say),either the red subgraph contains a copy of G or the blue subgraph contains H. Also set r(G) = r(G,G) ; these are called the diagonal Ramsey numbers. These definitions are taken from Chvátal and Harary [1J ; other terminology will follow Harary [2]. For a survey of known results concerning these generalized Ramsey numbers, see [3]. A natural question about Ramsey numbers is how small, or large, they can be. We make some definitions. If G is a set of graphs, define exr(G) by exr(G) = min r(G) ; GEG also if G and H are sets of graphs, set exr(G,H) = min r(G,TI). GEG HEH We also define Exr(G) and Exr(G,H) similarly, with min replaced by max. Note that exr(G,G) <_ exr (G). This inequality can be strict. In fact, theorem 4 .1 of [4] shows that exr(G,G)/exr(G) can be made arbitrarily small, even for sets G containing only two graphs. Likewise, theorem 2 .5 of [4] shows that Exr(G,G)/Fxr(G) can be made arbitrarily large. Define C n to be the set of connected graphs on n points, to be the set of graphs on n points with no isolates, and K n to UTILITAS MATHEMATICA Vol. 9 (1976), pp. 2 4 7-258 .