Polymath's combinatorial proof of the density Hales-Jewett theorem

TL;DR

This paper explains the combinatorial proof of the density Hales-Jewett theorem, which generalizes Szemerédi's theorem by showing large subsets of high-dimensional grids contain combinatorial lines, with implications for arithmetic progressions.

Contribution

It provides an exposition of Polymath's combinatorial proof of the density Hales-Jewett theorem, a key result in combinatorics and additive number theory.

Findings

The density Hales-Jewett theorem holds for sufficiently large dimensions.

Sets with positive density in high-dimensional grids contain combinatorial lines.

The theorem implies Szemerédi's theorem on arithmetic progressions.

Abstract

This is an exposition of the combinatorial proof of the density Hales--Jewett theorem, due to D.\,H.\,J. Polymath in 2012. The theorem says that for given $\de>0$ and $k$, for every $n>n_0$ every set $A\sus\{1,2,\ds,k\}^n$ with $|A|\ge\de k^n$ contains a combinatorial line. It implies Szemer\'edi's theorem, which claims that for given $\de>0$ and $k$, for every $n>n_0$ every set $A\sus\{1,2,\ds,n\}$ with $|A|\ge\de n$ contains a $k$-term arithmetic progression.