Shattering-extremal set systems from Sperner families

TL;DR

This paper investigates shattering-extremal set systems, proving a conjecture that such systems can be extended by one set while maintaining their extremal property, specifically within families derived from Sperner families.

Contribution

The paper proves a conjecture about the extendability of shattering-extremal set systems for a class derived from Sperner families.

Findings

Proves the conjecture for a class of set systems from Sperner families.

Shows that shattering-extremal families can be extended by one set.

Provides structural insights into shattering-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-Shelah lemma 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 \emph{shattering-extremal} if it shatters exactly $|\mathcal{F}|$ sets. A conjecture of R\'onyai and the second author and of Litman and Moran states that if a family is shattering-extremal then one can add a set to it and the resulting family is still shattering-extremal. Here we prove this conjecture for a class of set systems defined from Sperner families.