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

## TL;DR

This paper links the density Hales-Jewett number to communication complexity in the Number On the Forehead model, providing new insights and bounds, and suggesting that understanding this complexity can deepen knowledge of the Hales-Jewett theorem.

## Contribution

It establishes a correspondence between the density Hales-Jewett number and a communication complexity problem, revealing the tightness of previous bounds and proposing a new perspective on this combinatorial quantity.

## Key findings

- The density Hales-Jewett number equals the maximal size of a certain partition in a communication problem.
- The communication complexity of the problem matches the minimal partition size avoiding combinatorial lines.
- A new lower bound on the Hales-Jewett number is derived from a communication protocol.

## 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 $k \ge 3$ the density Hales-Jewett number $c_{n,k}$ is equal to the maximal size of a cylinder intersection in the problem $Part_{n,k}$ 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 $Part_{n,k}$, 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 \cite{chattopadhyay2007languages} on $Part_{n,k}$ 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 \cite{polymath2010moser} which is roughly $c_{n,k}/k^n \ge \exp(-O(\log n)^{1/\lceil \log_2 k\rceil})$. This lower bound follows from a protocol for $Part_{n,k}$. It is interesting to better understand the communication complexity of $Part_{n,k}$ 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.