Non-Ramsey Graphs Are c log n-Universal

We prove that for any c1>0 there exists c2>0 such that the following state- ment is true: If G is a graph with n vertices and with the property that neither G nor its complement contains a complete graph Kl, where l=c1logn then G is c2logn-universal, i.e., G contains all subgraphs with c2logn vertices as induced subgraphs.