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A slight modification of the proof of Szemerédi’s cube lemma gives that if a set $S \subset [1, n]$ satisfies $| S | \geq \frac{n}{2}$ , 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 $\Pr {s \in S} = \frac{1}{2}$ for $1 \leq s \leq n$ , 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.

Note on generating all subsets of a finite set with disjoint unions.

Ellis, David (2009)

The Electronic Journal of Combinatorics [electronic only]

Note on the gammoids arising from undirected graphs

Dănuţ Marcu (1990)

Rendiconti del Seminario Matematico della Università di Padova

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