Large cliques and independent sets all over the place

We study 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 \(2^{2^{(\log\log n)^{1/2+o(1)}}}\) obtaining a twofold sub-polynomial improvement over the upper bound of about \(\sqrt{n}\) 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 \(\log n\) 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.