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.

💡 Research Summary

The paper addresses a long‑standing open problem in multiparty communication complexity: establishing a strong lower bound for the set‑intersection (or disjointness) problem in the number‑on‑the‑forehead (NOF) model. In the k‑party NOF model each player sees all inputs except his own and communication proceeds on a shared blackboard. While the nondeterministic complexity of disjointness is only O(log n) (a prover can simply point to a common “‑1” coordinate), the best known lower bound for randomized protocols with k≥3 was merely Ω(log n/(k‑1)). The authors dramatically improve this bound, showing that any public‑coin randomized protocol with error ≤1/3 must use at least

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