A new proof of the density Hales-Jewett theorem

By None None. 2009

157 citations recorded by the source corpus.

The Hales-Jewett theorem asserts that for every r and every k there exists n such that every r-colouring of the n-dimensional 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 Hales-Jewett 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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/efe0378306c0acf5a5e883b30d8628d11e069f85

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

An Arithmetic Regularity Lemma, An Associated Counting Lemma, and Applications (2010, background): “c examples of this paradigm. In §6 we give a “new” proof of Szemer´edi’s famous theorem on arithmetic progressions. This is hardly exciting nowadays, with at least 14 proofs already in the literature [2, 3, 9, 12, 16, 17, 42, 43, 46, 48, 50] as well as (slightly implicitly) in [4, 33, 55]. However t”
A Probabilistic Technique for Finding Almost-Periods of Convolutions (2010, methodology): “However, only the Fourier-analytic proofs seem to have given decent bounds: the methods that have not used Fourier analysis have generally been accompanied by tower-type bounds, establishing only that r3(N) ≪ N/ logN ; see [44, Chapter 10] for references, as well as the more recent work [30].”
Learning and Smoothed Analysis (2009, unknown): “Interestingly, the distribution ρ1/2, a distribution with many appealing mathematical properties, has recently been used to simplify the proof of the density Hales-Jewett Theorem [11].”