On graphs with subgraphs having large independence numbers

Abstract

Let G be a graph on n vertices in which every induced subgraph on s = log3n 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) = O&Theta(log2n/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. © 2007 Wiley Periodicals, Inc.