Separating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem

Thomas Hayes
unpublished
For every positive integer k, we construct an explicit family of functions f : {0, 1} n → {0, 1} which has (k + 1)party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Ω n k 4 2 k under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely,
more » ... nkiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain Ks,t as a subgraph.
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