---
tag: J1VM4F8KZP
kind: statement
role: theorem
name: Density Hales-Jewett theorem
sources:
  - wikipedia: en:640714@r1368717634
    locator: Hales–Jewett theorem, section Connections with other theorems > Density Hales-Jewett theorem
    note: adopted
    mention: wikipedia://en/640714#99ZP
    sections: [Statement]
inventory: [A2QTQ2GQS5]
---

## Statement

Just as van der Waerden's theorem has a stronger *density version* in Szemerédi's theorem, the Hales–Jewett theorem also has a density version. In this strengthened version of the Hales–Jewett theorem, instead of coloring the entire hypercube *W*${}^{H}_{n}$ into *c* colors, one is given an arbitrary subset *A* of the hypercube *W*${}^{H}_{n}$ with some given density 0 \< δ \< 1. The theorem states that if *H* is sufficiently large depending on *n* and δ, then the set *A* must necessarily contain an entire combinatorial line.

The density Hales–Jewett theorem was originally proved by Furstenberg and Katznelson using ergodic theory. In 2009, the Polymath Project developed a new proof of the density Hales–Jewett theorem based on ideas from the proof of the corners theorem. Dodos, Kanellopoulos, and Tyros gave a simplified version of the Polymath proof.