Shattering-extremal set systems of small VC-dimension

TL;DR

This paper characterizes shattering-extremal set systems with VC-dimension 1 using inclusion graphs and explores their properties through projections, advancing understanding of extremal combinatorial structures.

Contribution

It provides a complete characterization of shattering-extremal set systems of VC-dimension 1 and links extremality to projections of bounded VC-dimension.

Findings

Characterization of shattering-extremal set systems of VC-dimension 1

Relation between extremality and set system projections

Use of inclusion graphs to describe extremal set systems

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.