Density Hales-Jewett theorem

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.