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 was Omega (log n/k-1). By results of Beame, Pitassi, and Segerlind, this implies 2^{n Omega(1)} lower bounds on the size of tree-like Lovasz-Schrijver proof systems needed to refute certain unsatisfiable CNFs, and super-polynomial lower bounds on the size of a broad class of tree-like proof systems whose terms are degree-d polynomial inequalities for d=log log n-O(log log log n). To prove our bound, we develop a new technique for showing lower bounds in the number-on-the-forehead model which is based on the norm induced by cylinder intersections. This bound naturally extends the linear program bound for rank useful in the two-party case to the case of more than two parties, where the fundamental concept of monochromatic rectangles is replaced by monochromatic cylinder intersections. Previously, the only general method known for showing lower bounds in the unrestricted number-on-the-forehead model was the discrepancy method, which is limited to bounds of size O(log n) for disjointness. To analyze the bound given by our new technique for the disjointness function, we build on an elegant framework developed by Sherstov in the two-party case and Chattopadhyay in the multi-party case which relates polynomial degree to communication complexity. Using this framework we are able to obtain bounds for any tensor of the form F(x₁,...,x_k)=f(x₁Lambda...Lambdax_k) where f is a function which only depends on the number of ones in the input.