Disjointness is Hard in the Multiparty Number-on-the-Forehead Model

Abstract

We show that disjointness requires randomized communication $$\Omega(\frac{n^{1/(k+1)}}{2^{2^k}})$$ in the general k-party number-on-the-forehead model of complexity. The previous best lower bound for k ≥ 3 was $$\frac{{\rm log} \, n}{k-1}$$ . 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.