More Shattering News

Abstract

An ordered variant of the well-known set theory concept of shattering was introduced by Anstee, Rónyai, and Sali. In this paper, we prove several new results related to order shattering. Given a family $\mathcal{F}$ of subsets of $[n]$, we show that $\mathrm{osh}(\mathcal{F})$, the family of all sets order shattered by $\mathcal{F}$, coincides with $T(\mathcal{F})$, the family obtained from $\mathcal{F}$ by the down-shift operation. We then give a full characterization of all sets that can be order shattered by some $\ell$-Sperner family. Finally, we completely determine $\mathrm{osh}\left(\left(\genfrac{}{}{0ex}{}{[n]}{a}\right)\cup \left(\genfrac{}{}{0ex}{}{[n]}{b}\right)\right)$.