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arXiv · 1211.0732

Shattering-extremal set systems of small VC-dimension

Tamás Mészáros·Lajos Rónyai
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Abstract

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S={F \cap S : F \in \mathcal{F}}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly $|\mathcal{F}|$ sets. We characterize shattering extremal set systems of Vapnik-Chervonenkis dimension 1 in terms of their inclusion graphs. Also from the perspective of extremality, we relate set systems of bounded Vapnik-Chervonenkis dimension to their projections.

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Tamás Mészáros, Lajos Rónyai. 2012-11-04. Shattering-extremal set systems of small VC-dimension. https://arxiv.org/abs/1211.0732

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