Let A be a finite set with , let n be a positive integer, and let $A^n$ denote the discrete $n\text {-dimensional}$ hypercube (that is, $A^n$ is the Cartesian product of n many copies of A). Given a family $\langle D_t:t\in A^n\rangle $ of measurable events in a probability space (a stochastic process), what structural information can be obtained assuming that the events $\langle D_t:t\in A^n\rangle $ are not behaving as if they were independent? We obtain an answer to this problem (in a strong quantitative sense) subject to a mild ‘stationarity’ condition. Our result has a number of combinatorial consequences, including a new (and the most informative so far) proof of the density Hales-Jewett theorem.

中文翻译：

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离散超立方体索引的随机过程的结构定理

让一种是一个有限集 ， 让n为正整数，令 $A^n$ 表示离散的 $n\text {维}$ 超立方体（即， $A^n$ 是的笛卡尔积n许多副本一种）。给定一个家庭 $\langle D_t:t\in A^n\rangle $ 概率空间中的可测量事件（随机过程），假设事件可以得到什么结构信息 $\langle D_t:t\in A^n\rangle $ 不表现得好像他们是独立的？我们在温和的“平稳性”条件下获得了这个问题的答案（在强烈的定量意义上）。我们的结果有许多组合结果，包括密度 Hales-Jewett 定理的新证明（也是迄今为止信息量最大的证明）。