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Polymath and the density Hales-Jewett theorem. (English) Zbl 1223.05305
Bárány, Imre (ed.) et al., An irregular mind. Szemerédi is 70. Dedicated to Endre Szemerédi on the occasion of his seventieth birthday. Berlin: Springer (ISBN 978-3-642-14443-1/pbk). Bolyai Society Mathematical Studies 21, 659-687 (2010).
Summary: 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 [“A density version of the Hales-Jewett theorem,” J. Anal. Math. 57, 64–119 (1991; Zbl 0770.05097)], H. Furstenberg and Y. 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.
For the entire collection see [Zbl 1196.00042].

05D10 Ramsey theory
Biographic References:
Polymath, D. H. J.