More Shattering News

TL;DR

This paper explores order shattering in set families, establishing its equivalence with down-shift operations, characterizing order shattering by Sperner families, and determining order shattered sets for specific unions.

Contribution

It proves the equivalence of order shattering with down-shift operations and characterizes order shattering in Sperner families.

Findings

Order shattering coincides with the down-shift operation.

Full characterization of order shattered sets by Sperner families.

Determination of order shattered sets for unions of two levels.

Abstract

An ordered variant of the well-known set theory concept of shattering was introduced by Anstee, R\'onyai, 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 $\mathrm{osh}\left(\binom{[n]}{a}\cup\binom{[n]}{b}\right)$.