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More Shattering News

Attila Sali, Jun Yan

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 $\mathrm{osh}\left(\binom{[n]}{a}\cup\binom{[n]}{b}\right)$.

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 of subsets of , we show that , the family of all sets order shattered by , coincides with , the family obtained from by the down-shift operation. We then give a full characterization of all sets that can be order shattered by some -Sperner family. Finally, we completely determine .
Paper Structure (6 sections, 23 theorems, 39 equations)

This paper contains 6 sections, 23 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Shifting
  3. $\ell$-Sperner families
  4. Two complete levels
  5. $d=2$
  6. $d>2$

Key Result

Proposition 1.4

A set $S\subseteq[n]$ with elements $s_1< s_2< \cdots < s_k$ is order shattered by $\mathcal{F}\subseteq\mathcal{P}([n])$ if and only if there exists a subfamily $\mathcal{G}=\{G_1,G_2,\ldots, G_{2^k}\}\subseteq\mathcal{F}$ of size $2^k$ such that the following hold.

Theorems & Definitions (54)

  • Definition 1.1
  • Definition 1.2: bollobas1989reversebollobas1995defect
  • Definition 1.3: anstee2002shattering
  • Proposition 1.4
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Claim 2.3
  • proof : Proof of Claim \ref{['claim']}
  • Definition 3.1
  • ...and 44 more