Inner Product and Set Disjointness: Beyond Logarithmically Many Parties

A basic goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems $f\colon(\{0,1\}^n)^{k}\to\{0,1\}$ with $k\gg\log n$ parties. We study the problems of inner product and set disjointness and determine their randomized communication complexity for every $k\geq\log n$, showing in both cases that $\Theta(1+\lceil\log n\rceil/\log\lceil1+k/\log n\rceil)$ bits are necessary and sufficient. In particular, these problems admit constant-cost protocols if and only if the number of parties is $k\geq n^{\epsilon}$ for some constant $\epsilon>0.$