# Polymath and The Density Hales-Jewett Theorem

• W. T. Gowers
Chapter
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 21)

## Abstract

Van der Waerden’s theorem has two well-known and very different generalizations. One is the Hales-Jewett theorem, one of the cornerstones of Ramsey theory. The other is Endre Szemerédi’s famous density version of the theorem, which has played a pivotal role in the recent growth of additive combinatorics. In 1991 Furstenberg and Katznelson proved the density Hales-Jewett theorem, a result that has the same relationship to the Hales-Jewett theorem that Szemerédi’s theorem has to van der Waerden’s theorem. Furstenberg and Katznelson used a development of the ergodic-theoretic machinery introduced by Furstenberg. Very recently, a new and much more elementary proof was discovered in a rather unusual way - by a collaborative process carried out in the open with the help of blogs and a wiki. In this informal paper, we briefly discuss this discovery process and then give a detailed sketch of the new proof.

## Keywords

Proper Subset Arithmetic Progression Uniform Measure Average Argument Combinatorial Line
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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