Advertisement

Polymath and The Density Hales-Jewett Theorem

  • W. T. GowersEmail author
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Ajtai and E. Szemerédi, Sets of lattice points that form no squares, Stud. Sci. Math. Hungar., 9 (1974), 9–11.MathSciNetGoogle Scholar
  2. [2]
    T. Austin, Deducing the density Hales-Jewett theorem from an infinitary removal lemma, http://arxiv.org/abs/0903.1633, 2009.
  3. [3]
    V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden’s and Szemerédi’s theorems, Jour. AMS, 9 (1996), 725–753.zbMATHMathSciNetGoogle Scholar
  4. [4]
    H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. d’Analyse Math., 31 (1977), 204–256.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. d’Analyse Math., 34 (1978), 275–291.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    H. Furstenberg and Y. Katznelson, A density version of the Hales-Jewett Theorem, J. d’Analyse Math., 57 (1991), 64–119.zbMATHMathSciNetGoogle Scholar
  7. [7]
    A. W. Hales and R. I. Jewett, Regularity and positional games, Trans. AMS, 106(2) (1963), 222–229.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    D. H. J. Polymath, A new proof of the density Hales-Jewett theorem, http://arxiv.org/abs/0910.3926
  9. [9]
    D. H. J. Polymath, Density Hales-Jewett and Moser numbers, this volume.Google Scholar
  10. [10]
    I. Z. Ruzsa and E. Szemerédi, Triple systems with no six points carrying three triangles, Colloq. Math. Soc. J. Bolyai, 18 (1978), 939–945.Google Scholar
  11. [11]
    S. Shelah, Primitive recursive bounds for van der Waerden numbers, Jour. AMS, 1:683–697, 1988.zbMATHGoogle Scholar
  12. [12]
    J. Solymosi, A note on a question of Erdös and Graham, Combin. Probab. Comput., 13(2) (2004), 263–267.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith., 27 (1975), 199–245.-zbMATHMathSciNetGoogle Scholar
  14. [14]
    B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Archief voor Wiskunde, 15 (1927), 212–216.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag 2010

Authors and Affiliations

  1. 1.University of Cambridge Department of Pure Mathematics and Mathematical StatisticsCambridgeUK

Personalised recommendations