A Simple Proof of the Density Hales–Jewett Theorem

Dodos, Pandelis; Kanellopoulos, Vassilis; Tyros, Konstantinos

doi:10.1093/imrn/rnt041

Cited by 16 publications

(21 citation statements)

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“…, a n−1 ∈ A where n is a positive integer and A is a finite alphabet (i.e., a finite set) with at least two letters. The building blocks of the corresponding semirings were introduced by Shelah [20] in his work on the Hales-Jewett numbers, and are essential tools in all known combinatorial proofs of the density Hales-Jewett theorem (see [7,17,24]).…”

Section: Definition 21mentioning

confidence: 99%

Szemerédi's Regularity Lemma via Martingales

Dodos

Kanellopoulos

Karageorgos

2016

Electron. J. Combin.

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Section: Definition 21mentioning

confidence: 99%

Szemerédi's Regularity Lemma via Martingales

Dodos

Kanellopoulos

Karageorgos

2016

Electron. J. Combin.

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“…, k} n as base-k representations of integers, and then every combinatorial line is an arithmetic progression of length k (but not vice versa). Recently, this proof has been simplified yet further [8].…”

Section: 1mentioning

confidence: 99%

Erdős and Arithmetic Progressions

Gowers

2013

Bolyai Society Mathematical Studies

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“…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 .…”

Section: Introductionmentioning

confidence: 99%

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

Shraibman

2018

Information Processing Letters

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For integers n and k, the density Hales-Jewett number c n,k is defined as the maximal size of a subset of [k] n that contains no combinatorial line. We show that for k ≥ 3 the density Hales-Jewett number c n,k is equal to the maximal size of a cylinder intersection in the problem P art n,k of testing whether k subsets of [n] form a partition. It follows that the communication complexity, in the Number On the Forehead (NOF) model, of P art n,k , is equal to the minimal size of a partition of [k] n into subsets that do not contain a combinatorial line. Thus, the bound in [7] on P art n,k using the Hales-Jewett theorem is in fact tight, and the density Hales-Jewett number can be thought of as a quantity in communication complexity. This gives a new angle to this well studied quantity.As a simple application we prove a lower bound on c n,k , similar to the lower bound in [19] which is roughly c n,k /k n ≥ exp(−O(log n) 1/⌈log 2 k⌉ ). This lower bound follows from a protocol for P art n,k . It is interesting to better understand the communication complexity of P art n,k as this will also lead to the better understanding of the Hales-Jewett number. The main purpose of this note is to motivate this study.

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A Simple Proof of the Density Hales–Jewett Theorem

Cited by 16 publications

References 15 publications

Szemerédi's Regularity Lemma via Martingales

Szemerédi's Regularity Lemma via Martingales

Erdős and Arithmetic Progressions

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

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