Large cliques and independent sets all over the place

We consider the following question raised by Erdős 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 log n? We construct examples showing that m is at most 2^(2^((loglog n)^(1/2+o(1)))) obtaining a twofold sub-polynomial improvement over the upper bound of about square root of 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 homogeneous subsets contain both cliques and independent sets of size log n 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. This is joint work with Noga Alon and Benny Sudakov.