## Non-degenerate Hilbert cubes in random sets.(English)Zbl 1126.11014

A $$k$$-cube $$H\subseteq\{1,\ldots,n\}$$ has the form $$\{a_0+\sum_{i\in I}a_i: I\subseteq\{1,\dots,k\}\}$$ with $$a_0,a_1,\ldots,a_k\in\{1,\dots,n\}$$. It is called non-degenerate if $$| H| =2^k$$. By D. S. Gunderson and V. Rödl’s refinement [Comb. Probab. Comput. 7, 65–79 (1998; Zbl 0892.05050)] of Szemerédi’s cube lemma, each set $$S\subseteq\{1,\ldots,n\}$$ with $$| S| \geq n/2$$ contains a $$\lfloor\log_2\log_2n-3\rfloor$$-cube.
In the paper under review, the author shows that in a random set $$S$$ with Pr$$(s\in S)=1/2$$ for all $$s=1,\dots,n$$, $$\max\{k:S$$ contains a non-degenerate $$k$$-cube} is almost everywhere nearly $$\log_2\log_2n+\log_2\log_2\log_2n$$.

### MathOverflow Questions:

Lower bounds for the density variant of the Hilbert cube problem

### MSC:

 11B75 Other combinatorial number theory 05D40 Probabilistic methods in extremal combinatorics, including polynomial methods (combinatorial Nullstellensatz, etc.) 11P99 Additive number theory; partitions

### Keywords:

Hilbert cube; subset sum; random set

Zbl 0892.05050
Full Text:

### References:

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