### Abstract

We offer a new proof of Furstenberg and Katznelson's density version of the Hales-Jewett Theorem: Theorem For any δ > 0 there is some N_{0} ≥ 1 such that whenever A ⊆ [k]^{N} with N ≥ N_{0} and {pipe}A{pipe} ≥ δk^{N}, A contains a combinatorial line, that is, for some I ⊆ [N] nonempty and w_{0} ∈ [k]^{[N]\I} we have A ⊇ {w : w{pipe}_{[N]\I} = w_{0}, w{pipe}I = const.}. Following Furstenberg and Katznelson, we first show that this result is equivalent to a 'multiple recurrence' assertion for a class of probability measures enjoying a certain kind of stationarity. However, we then give a quite different proof of this latter assertion through a reduction to an infinitary removal lemma in the spirit of Tao (J. Anal. Math. 103, 1-45, 2007) (and also its recent re-interpretation in (J. Anal. Math., to appear)). This reduction is based on a structural analysis of these stationary laws closely analogous to the classical representation theorems for various partially exchangeable stochastic processes in the sense of Hoover (Relations on probability spaces and arrays of random variables, 1979), Aldous (in Exchangeability in Probability and Statistics, 165-170, 1982; Lecture Notes in Math. 1117, 1-198, 1985) and Kallenberg (J. Theor. Probab. 5(4), 727-765, 1992). However, the underlying combinatorial arguments used to prove this theorem are rather different from those required to work with exchangeable arrays, and involve crucially an observation that arose during ongoing work by a collaborative team of authors (http://gowers.wordpress.com/) to give a purely finitary proof of the above theorem.

Original language | English (US) |
---|---|

Pages (from-to) | 615-633 |

Number of pages | 19 |

Journal | Journal of Theoretical Probability |

Volume | 24 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2011 |

### Fingerprint

### Keywords

- Density Hales-Jewett
- Ergodic Ramsey theory
- Exchangeability

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Theoretical Probability*,

*24*(3), 615-633. https://doi.org/10.1007/s10959-011-0373-4

**Deducing the Density Hales-Jewett Theorem from an Infinitary Removal Lemma.** / Austin, Tim.

Research output: Contribution to journal › Article

*Journal of Theoretical Probability*, vol. 24, no. 3, pp. 615-633. https://doi.org/10.1007/s10959-011-0373-4

}

TY - JOUR

T1 - Deducing the Density Hales-Jewett Theorem from an Infinitary Removal Lemma

AU - Austin, Tim

PY - 2011/9

Y1 - 2011/9

N2 - We offer a new proof of Furstenberg and Katznelson's density version of the Hales-Jewett Theorem: Theorem For any δ > 0 there is some N0 ≥ 1 such that whenever A ⊆ [k]N with N ≥ N0 and {pipe}A{pipe} ≥ δkN, A contains a combinatorial line, that is, for some I ⊆ [N] nonempty and w0 ∈ [k][N]\I we have A ⊇ {w : w{pipe}[N]\I = w0, w{pipe}I = const.}. Following Furstenberg and Katznelson, we first show that this result is equivalent to a 'multiple recurrence' assertion for a class of probability measures enjoying a certain kind of stationarity. However, we then give a quite different proof of this latter assertion through a reduction to an infinitary removal lemma in the spirit of Tao (J. Anal. Math. 103, 1-45, 2007) (and also its recent re-interpretation in (J. Anal. Math., to appear)). This reduction is based on a structural analysis of these stationary laws closely analogous to the classical representation theorems for various partially exchangeable stochastic processes in the sense of Hoover (Relations on probability spaces and arrays of random variables, 1979), Aldous (in Exchangeability in Probability and Statistics, 165-170, 1982; Lecture Notes in Math. 1117, 1-198, 1985) and Kallenberg (J. Theor. Probab. 5(4), 727-765, 1992). However, the underlying combinatorial arguments used to prove this theorem are rather different from those required to work with exchangeable arrays, and involve crucially an observation that arose during ongoing work by a collaborative team of authors (http://gowers.wordpress.com/) to give a purely finitary proof of the above theorem.

AB - We offer a new proof of Furstenberg and Katznelson's density version of the Hales-Jewett Theorem: Theorem For any δ > 0 there is some N0 ≥ 1 such that whenever A ⊆ [k]N with N ≥ N0 and {pipe}A{pipe} ≥ δkN, A contains a combinatorial line, that is, for some I ⊆ [N] nonempty and w0 ∈ [k][N]\I we have A ⊇ {w : w{pipe}[N]\I = w0, w{pipe}I = const.}. Following Furstenberg and Katznelson, we first show that this result is equivalent to a 'multiple recurrence' assertion for a class of probability measures enjoying a certain kind of stationarity. However, we then give a quite different proof of this latter assertion through a reduction to an infinitary removal lemma in the spirit of Tao (J. Anal. Math. 103, 1-45, 2007) (and also its recent re-interpretation in (J. Anal. Math., to appear)). This reduction is based on a structural analysis of these stationary laws closely analogous to the classical representation theorems for various partially exchangeable stochastic processes in the sense of Hoover (Relations on probability spaces and arrays of random variables, 1979), Aldous (in Exchangeability in Probability and Statistics, 165-170, 1982; Lecture Notes in Math. 1117, 1-198, 1985) and Kallenberg (J. Theor. Probab. 5(4), 727-765, 1992). However, the underlying combinatorial arguments used to prove this theorem are rather different from those required to work with exchangeable arrays, and involve crucially an observation that arose during ongoing work by a collaborative team of authors (http://gowers.wordpress.com/) to give a purely finitary proof of the above theorem.

KW - Density Hales-Jewett

KW - Ergodic Ramsey theory

KW - Exchangeability

UR - http://www.scopus.com/inward/record.url?scp=79960841149&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960841149&partnerID=8YFLogxK

U2 - 10.1007/s10959-011-0373-4

DO - 10.1007/s10959-011-0373-4

M3 - Article

VL - 24

SP - 615

EP - 633

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 3

ER -