On graphs with subgraphs having large independence numbers

Let G be a graph on n vertices in which every induced subgraph on s = log n vertices has an independent set of size at least t = log n. What is the largest q = q(n) so that every such G must contain an independent set of size at least q ? This is one of several related questions raised by Erdős and Hajnal. We show that q(n) = Θ(log n/ log log n), investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey-type problem.