A new proof of the density Hales-Jewett theorem

Abstract

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 Erdős and Turán in 1936, proved by Szemerédi 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 Szemerédi'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 δ contains a combinatorial line if n is at least as big as a tower of 2s of height O(1/δ 2 ). Our proof is surprisingly simple: indeed, it gives arguably the simplest known proof of Szemerédi's theorem.

An analogue of the Erdős-Stone theorem for finite geometries

“…The value of R 3.1 (m, q, c, ǫ) provided by Theorem 3.1 depends on that of R 2.2 (m, q, ǫ), for which the bounds in [8] are Ackermann-like for all q > 2. In the binary case, however, the main theorem of [1] implies that the relatively small function R 2.2 (m, 2, ǫ) = 2 m−2 ⌈1 − log 2 ǫ⌉ will satisfy Theorem 2.2.…”

“…This result can be obtained as an easy application of the density version of the multidimensional Hales-Jewett theorem, also proved by Furstenberg and Katznelson [6], in 1991, using ergodic theory. An easier proof was later obtained via the polymath project [8]. The "easier proof" is still, however, more than 30 pages long.…”

The densest matroids in minor-closed classes with exponential growth rate

“…Computability of f 2.1 follows from computability of the functions defined in ([2]: Lemmas 3.3, 4.3, 5.1 and 5.3). Computability of f 2.2 relies on computability of the functions in ([5]: Lemmas 4.3 and 5.2, Theorem 6.1) which themselves rely on computability of f 2.1 , as well as that of the function defined in ( [4], Lemma 8.1) and the main result of [7], the Density Hales-Jewett theorem. Checking the computability of all these functions directly is straightforward except for the theorem in [7], and in that theorem the authors take care to provide computable upper bounds for the associated function (see [7], Theorem 1.5).…”

A note on multiparty communication complexity and the Hales–Jewett theorem

“…The proof of Furstenberg and Katznelson used ergodic-theory and gave no explicit bound on c n,k . Recently, additional proofs of this theorem were found [18,2,8]. The proof of [18] is the first combinatorial proof of the density Hales-Jewett theorem, and also provides effective bounds for c n,k .…”