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 ErdAs conjectured that r(K (3),Q (n) )=2 (n+1)-1 for every naa"center dot, 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) )a (c) 1/27000 center dot 2 (n) . Here we show that r(K (3),Q (n) )=(1+o(1))2 (n+1) as n -> a.