On the Ramsey number of the triangle and the cube

The Ramsey number r ( K 3 , Q n ) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K N contains either a red n -dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erdős conjectured that r ( K 3 , Q n )=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 ( K 3 , Q n )⩽7000·2 n . Here we show that r ( K 3 , Q n )=(1+ o (1))2 n +1 as n →∞.