The Ramsey number of the clique and the hypercube

TL;DR

This paper determines the exact Ramsey number for a clique versus a hypercube in large graphs, confirming a longstanding conjecture and improving previous bounds.

Contribution

The authors prove the exact value of r(K_s,Q_n) for all s and large n, resolving a question posed in 1983 and advancing understanding of hypercube and clique colorings.

Findings

Exact formula for r(K_s,Q_n) for large n

Confirmed a 1983 conjecture by Burr and Erdős

Improved previous bounds on hypercube-clique Ramsey numbers

Abstract

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.