Disjointness is hard in the multi-party number on the forehead model

We show that disjointness requires randomized communication Omega(n^{1/(k+1)}/2^{2^k}) in the general k-party number-on-the-forehead model of complexity. The previous best lower bound for k >= 3 was log(n)/(k-1). Our results give a separation between nondeterministic and randomized multiparty number-on-the-forehead communication complexity for up to k=log log n - O(log log log n) many players. Also by a reduction of Beame, Pitassi, and Segerlind, these results imply subexponential lower bounds on the size of proofs needed to refute certain unsatisfiable CNFs in a broad class of proof systems, including tree-like Lovasz-Schrijver proofs.