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The research presented in this paper was motivated by our aim to study a problem due to J. Bourgain [3]. The problem in question concerns the uniform bound-edness of the classical separation rank of the elements of a separable compact set of the first Baire class. In the sequel we shall refer to these sets (separable or non-separable) as Rosenthal compacta… (More)

We prove a variant of the abstract probabilistic version of Szemerédi's regularity lemma, due to Tao, which applies to a number of structures (including graphs, hy-pergraphs, hypercubes, graphons, and many more) and works for random variables in L p for any p > 1. Our approach is based on martingale difference sequences.

For every integer k 2 let [k] <N be the set of all words over k. A Carlson–Simpson tree of [k] <N of dimension m 1 is a subset of [k] <N of the form {w} ∪ w w 0 (a 0) ... where w is a word over k and (wn) m−1 n=0 is a finite sequence of left variable words over k. We study the behavior of a family of measurable events in a probability space indexed by the… (More)

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