A new proof of the density HalesJewett theorem
Abstract
The HalesJewett theorem asserts that for every r and every k there exists n such that every rcolouring of the ndimensional grid {1,...,k}^n contains a combinatorial line. This result is a generalization of van der Waerden's theorem, and it is one of the fundamental results of Ramsey theory. The theorem of van der Waerden has a famous density version, conjectured by Erdos and Turan in 1936, proved by Szemeredi in 1975, and given a different proof by Furstenberg in 1977. The HalesJewett theorem has a density version as well, proved by Furstenberg and Katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by Furstenberg in his proof of Szemeredi's theorem. In this paper, we give the first elementary proof of the theorem of Furstenberg and Katznelson, and the first to provide a quantitative bound on how large n needs to be. In particular, we show that a subset of {1,2,3}^n of density delta contains a combinatorial line if n is at least a tower of 2's of height O(1/delta^3). Our proof is reasonably simple: indeed, it gives what is arguably the simplest known proof of Szemeredi's theorem.
 Publication:

arXiv eprints
 Pub Date:
 October 2009
 arXiv:
 arXiv:0910.3926
 Bibcode:
 2009arXiv0910.3926P
 Keywords:

 Mathematics  Combinatorics;
 05D10
 EPrint:
 See also http://michaelnielsen.org/polymath1/index.php?title=Polymath1