Shattering-extremal set systems of VC dimension at most 2

TL;DR

This paper characterizes shattering-extremal set systems of VC dimension 2 using their inclusion graphs and addresses an open question about element removal in such systems.

Contribution

It provides a complete characterization of shattering-extremal set systems of VC dimension 2 and solves an open problem regarding element exclusion.

Findings

Characterization of shattering-extremal set systems of VC dimension 2

Answer to an open question about element removal from these systems

Insights into the structure of 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. In this paper we characterize shattering-extremal set systems of Vapnik-Chervonenkis dimension $2$ in terms of their inclusion graphs, and as a corollary we answer an open question from \cite{VC1} about leaving out elements from shattering-extremal set systems in the case of families of Vapnik-Chervonenkis dimension $2$.