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Non-degenerate Hilbert cubes in random sets

Csaba Sándor — 2007

Journal de Théorie des Nombres de Bordeaux

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

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