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Csaba Sándor
Non-degenerate Hilbert cubes in random sets
Journal de théorie des nombres de Bordeaux, 19 no. 1 (2007), p. 249-261, doi: 10.5802/jtnb.585
Article PDF | Reviews MR 2332065 | Zbl 1126.11014

Résumé - Abstract

A slight modification of the proof of Szemerédi’s cube lemma gives that if a set $S\subset [1,n]$ satisfies $|S|\ge \frac{n}{2}$, then $S$ must contain a non-degenerate Hilbert cube of dimension $\lfloor \log _2\log _2n -3\rfloor $. In this paper we prove that in a random set $S$ determined by $\textrm{Pr}\lbrace s\in S\rbrace =\frac{1}{2}$ for $1\le s\le n$, the maximal dimension of non-degenerate Hilbert cubes is a.e. nearly $\log _2\log _2n+\log _2\log _2\log _2n$ and determine the threshold function for a non-degenerate $k$-cube.

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