We study the following question raised by Erd\H{o}s and Hajnal in the early
90's. Over all n-vertex graphs G what is the smallest possible value of m
for which any m vertices of G contain both a clique and an independent set
of size logn? We construct examples showing that m is at most
22(loglogn)1/2+o(1) obtaining a twofold sub-polynomial
improvement over the upper bound of about n coming from the natural
guess, the random graph. Our (probabilistic) construction gives rise to new
examples of Ramsey graphs, which while having no very large homogenous subsets
contain both cliques and independent sets of size logn in any small subset
of vertices. This is very far from being true in random graphs. Our proofs are
based on an interplay between taking lexicographic products and using
randomness.Comment: 12 page