  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.
