由无穷移引理推导密度Hales-Jewett定理
Deducing the Density Hales-Jewett Theorem from an infinitary removal lemma
摘要：
我们给出了Furstenberg和Katznelson的Hales-Jewett定理的密度版本的新证明：对于任何$δ> 0$，存在一些$N_0 \geq 1$，使得每当$A \subseteq [k]^N$具有$N \geq N_0$和$|A|\geq δk^N$时，$A$包含一个\textbf{combinatorial line}：即，对于一些非空的$I \subseteq [N]$和$w_0 \in [k]^{[N]\setminus I}$，我们有一个\supseteq \{w：w|_{[N] \setminus I} = w_0，w|_I = \rm{const.} \}。继Furstenberg和Katznelson，我们首先表明，这一结果是等价于一个“多重递归”断言一类概率措施享有一定的平稳性。然而，我们然后给一个完全不同的证明后一个断言通过减少到一个无穷删除引理的精神陶的工作无限随机超图（也是其最近重新解释的一个新的证明多维Szemeredi定理由本作者）。这种减少是基于结构分析这些平稳的法律密切类似的经典表示定理的各种部分可交换的随机过程的意义上的胡佛，奥尔德斯和Kallenberg。然而，用于证明这个定理的基本组合参数与可交换阵列所需的参数有很大的不同，并且关键地涉及到一个由作者组成的协作团队在进行中的工作中产生的观察，以给出上述定理的纯粹有限性证明。
英文摘要：
We offer a new proof of Furstenberg and Katznelson's density version of the Hales-Jewett Theorem: For any $δ> 0$ there is some $N_0 \geq 1$ such that whenever $A \subseteq [k]^N$ with $N \geq N_0$ and $|A|\geq δk^N$, $A$ contains a \textbf{combinatorial line}: that is, for some $I \subseteq [N]$ nonempty and $w_0 \in [k]^{[N]\setminus I}$ we have A \supseteq \{w: w|_{[N]\setminus I} = w_0, w|_I = \rm{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's work on infinite random hypergraphs (and also its recent re-interpretation in a new proof of the multidimensional Szemeredi Theorem by the present author). This reduction is based on a structural analysis of these stationary laws closely analogous to the classical representation theorems for various partial exchangeable stochastic processes in the sense of Hoover, Aldous and Kallenberg. 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 to give a purely finitary proof of the above theorem.
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