Ramsey numbers of cubes versus cliques

Abstract

The cube graph Q[subscript n] is the skeleton of the n-dimensional cube. It is an n-regular graph on 2[superscript n] vertices. The Ramsey number r(Q[subscript n] ;K[subscript s]) is the minimum N such that every graph of order N contains the cube graph Q[subscript n] or an independent set of order s. In 1983, Burr and Erdős asked whether the simple lower bound r(Q[subscript n] ;K[subscript s] )≥(s−1)(2[superscript n] −1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.