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  "body": "\n## Statement\n\nJust 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 \\\u003c δ \\\u003c 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.\n\nThe 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.\n",
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