On graphs with subgraphs having large independence numbers

Let G be a graph on n vertices in which every induced subgraph on s = log(3) 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 the several related questions raised by Erdos and Hajnal. We show that q(n) = Theta(log(2) 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. (C) 2007 Wiley Periodicals, Inc.