# Explicit separations between randomized and deterministic Number-on-Forehead communication

Symposium on the Theory of Computing（2023）

Abstract

We study the power of randomness in the Number-on-Forehead (NOF) model in
communication complexity. We construct an explicit 3-player function f:[N]^3
→{0,1}, such that: (i) there exist a randomized NOF protocol computing it
that sends a constant number of bits; but (ii) any deterministic or
nondeterministic NOF protocol computing it requires sending about (log
N)^1/3 many bits. This exponentially improves upon the previously best-known
such separation. At the core of our proof is an extension of a recent result of
the first and third authors on sets of integers without 3-term arithmetic
progressions into a non-arithmetic setting.

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