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

Abstract

For integers n and k, the density Hales-Jewett number c n,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 c n,k is equal to the maximal size of a cylinder intersection in the problem P art n,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 P art n,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 P art n,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 c n,k , similar to the lower bound in [19] which is roughly c n,k /k n ≥ exp(−O(log n) 1/⌈log 2 k⌉ ). This lower bound follows from a protocol for P art n,k . It is interesting to better understand the communication complexity of P art n,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.

Larger Corner-Free Sets from Combinatorial Degenerations

“…The NOF model is very rich in terms of connections to Ramsey theory and additive combinatorics [9,45,34,35,36], as well as applications to boolean models of compution such as branching programs and boolean circuits [15,10]. The goal in the NOF model is for k players to compute a fixed given function F : X 1 × • • • × X k → {0, 1} on inputs (x 1 , .…”

“…This paper is about constructing special combinatorial objects, namely "corner-free sets" in F n p × F n p , motivated (besides their inherent interest) by central problems in communication complexity, specifically in the study of the number on the forehead (NOF) model of communication introduced by Chandra, Furst and Lipton [15]. There has been much interest in (and progress on) the corner problem, variations of the problem, and connections to NOF communication, in particular in the recent works of Shraibman [45], Linial, Pitassi and Shraibman [34], Viola [50], Alon and Shraibman [5], and Linial and Shraibman [35,36]. In the recent work of Linial and Shraibman [35] a construction of large corner-free sets in [N ] × [N ] was obtained in an elegant manner by designing efficient NOF communication protocols for a specific communication problem (much like the upcoming Eval problem).…”