Non - degenerate Hilbert cubes in random sets par
نویسنده
چکیده
A slight modification of the proof of Szemerédi’s cube lemma gives that if a set S ⊂ [1, n] satisfies |S| ≥ n2 , then S must contain a non-degenerate Hilbert cube of dimension blog2 log2 n− 3c. 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 nondegenerate Hilbert cubes is a.e. nearly log2 log2 n+log2 log2 log2 n and determine the threshold function for a non-degenerate k-cube.
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