The Ramsey number of the clique and the hypercube

Gonzalo Fiz Pontiveros, Simon Griffiths, Robert Morris, David Saxton, Jozef Skokan

Published 2013-06-03Version 1

The Ramsey number r(K_s,Q_n) is the smallest positive integer N such that every red-blue colouring of the edges of the complete graph K_N on N vertices contains either a red n-dimensional hypercube, or a blue clique on s vertices. Answering a question of Burr and Erd\H{o}s from 1983, and improving on recent results of Conlon, Fox, Lee and Sudakov, and of the current authors, we show that r(K_s,Q_n) = (s-1) (2^n - 1) + 1 for every s \in \N and every sufficiently large n \in \N.