On the Ramsey number of the triangle and the cube

The Ramsey number r(K3, Qn) is the smallest integer N such that every redblue colouring of the edges of the complete graph KN contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erdős conjectured that r(K3, Qn) = 2 n+1 − 1 for every n ∈ N, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K3, Qn) 6 7000 · 2. Here we show that r(K3, Qn) = ( 1 + o(1) ) 2 as n→∞.