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 \(K_p\) 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 \(l\) points respectively and have point independence numbers of \(i\) and \(j\) respectively. Then \(N-1 \leq r(mG,nH) \leq N+C\), where \(N=km+ln- \min (mi,nj)\) 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, \(r(mK_3,nK_3)=3m+2n\) when \(m \geq n\), \(m \geq 2\).