More Shattering News

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 $\mathrm{o}\mathrm{s}\mathrm{h}(\mathcal{F})$, 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 $\mathrm{\ell }$-Sperner family. Finally, we completely determine $\mathrm{o}\mathrm{s}\mathrm{h}(\left(\genfrac{}{}{0px}{}{[n]}{a}\right)\cup \left(\genfrac{}{}{0px}{}{[n]}{b}\right))$.