A note on multiparty communication complexity and the Hales–Jewett theorem

Abstract

For integers n and k, the density Hales–Jewett numbercn,k is defined as the maximal size of a subset of [k]n that contains no combinatorial line. We show that for k≥3 the density Hales–Jewett number cn,k is equal to the maximal size of a cylinder intersection in the problem Partn,k of testing whether k subsets of [n] form a partition. It follows that the communication complexity, in the Number On the Forehead (NOF) model, of Partn,k, is equal to the minimal size of a partition of [k]n into subsets that do not contain a combinatorial line. Thus, the bound in [7] on Partn,k using the Hales–Jewett theorem is in fact tight, and the density Hales–Jewett number can be thought of as a quantity in communication complexity. This gives a new angle to this well studied quantity.As a simple application we prove a lower bound on cn,k, similar to the lower bound in [19] which is roughly cn,k/kn≥exp⁡(−O(log⁡n)1/⌈log2⁡k⌉). This lower bound follows from a protocol for Partn,k. It is interesting to better understand the communication complexity of Partn,k as this will also lead to the better understanding of the Hales–Jewett number. The main purpose of this note is to motivate this study.