A Note on Multiparty Communication Complexity and the Hales-Jewett Theorem

For integers n and k, the density Hales-Jewett number cn,k is defined as the maximal size of a subset of [k] that contains no combinatorial line. We prove a lower bound on cn,k, similar to the lower bound in [16], but with better dependency on k. The bound in [16] is roughly cn,k/k n ≥ exp(−O(log n)2 ) and we show cn,k/k n ≥ exp(−O(log n)). The proof of the bound uses the well-known construction of Behrend [3] and Rankin [17], as the one in [16], but does not require the recent refinements [7, 11, 14]. Instead our proof relies on an argument from communication complexity. In addition, 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] into subsets that do not contain a combinatorial line. Thus, Tesson’s bound on the problem using the Hales-Jewett theorem [19] is in fact tight, and the density Hales-Jewett number can be thought of as a quantity in communication complexity.