A remark on the Ramsey number of the hypercube

TL;DR

This paper improves the upper bound on the Ramsey number of the hypercube, showing it grows more slowly than previously thought, which advances understanding of hypercube colorings in combinatorics.

Contribution

The authors establish a tighter upper bound on the hypercube's Ramsey number, reducing it from exponential in 2^{2n} to a smaller exponential growth rate.

Findings

New upper bound: r(Q_n)=O(2^{2n - c n}) for some constant c>0

Improvement over previous bound: r(Q_n)=O(2^{2n})

Progress towards the Burr and Erdos conjecture

Abstract

A well known conjecture of Burr and Erdos asserts that the Ramsey number $r(Q_n)$ of the hypercube $Q_n$ on $2^n$ vertices is of the order $O(2^n)$. In this paper, we show that $r(Q_n)=O(2^{2n-c n})$ for a universal constant $c>0$, improving upon the previous best known bound $r(Q_n)=O(2^{2n})$, due to Conlon, Fox and Sudakov.