On a Ramsey-type problem of Erdős and Pach

Abstract

Erdős and Pach (1983) asked if there is some constant C>0 such that for any graph G on Ckln⁡k vertices either G or its complement G¯ has an induced subgraph on k vertices with minimum degree at least 12(k−1). They showed that the above statement holds with Ck2 in place of Ckln⁡k but that it does not hold with Ckln⁡k/ln⁡ln⁡k. We show that it holds with Ckln2⁡k, answering their question up to a ln⁡k factor.