An upper bound for the Ramsey numbers r ( & , G ) *

The Ramsey number r(H, G) is defined as the minimum N such that for any coloring of the edges of the N-vertex complete graph KN in red and blue, it must contain either a ted H or a blue G. In this paper we show that for any graph G without isolated vertices, r(K,, G)< 2qf 1 where G has q edges. In other words, any graph on 2q+ 1 vertices with independence number at most 2 contains every (isolate-free) graph on q edges. This establishes a 1980 conjecture of Harary. The result is best possible as a function of q.