Deducing the Density Hales–Jewett Theorem from an Infinitary Removal Lemma

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 |A|≥δk ^N, A contains a combinatorial line, that is, for some I⊆[N] nonempty and w ₀∈[k]^[N]∖I we have

$$A \supseteq \{w:\ w|_{[N]\setminus I} = w_0,\,w|_I = \mathrm{const.}\}.$$

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.