Generalized progressions in the integer Heisenberg group have VC-dimension at most 267, and in the free group on k generators at most 3k-1.
The Vapnik-Chervonenkis dimension of cubes in $\mathbb{R}^d$
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The Vapnik-Chervonenkis (VC) dimension of a collection of subsets of a set is an important combinatorial concept in settings such as discrete geometry and machine learning. In this paper we prove that the VC dimension of the family of $d$-dimensional cubes in $\mathbb R^d$ is $\lfloor(3d+1)/2\rfloor$.
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VC-dimension of generalized progressions in some nonabelian groups
Generalized progressions in the integer Heisenberg group have VC-dimension at most 267, and in the free group on k generators at most 3k-1.