Containments in families with forbidden subposets

TL;DR

This paper investigates the maximum number of containment pairs in set families avoiding certain height-2 posets, providing bounds, exact values for specific posets, and asymptotic results for path-like posets.

Contribution

It establishes upper bounds for containment pairs in families avoiding height-2 posets and determines exact and asymptotic values for specific posets.

Findings

Maximum pairs are O(n * binomial(n, n/2)) for any height-2 poset.

Exact maximum number of pairs found for butterfly and N posets.

Asymptotic behavior characterized for path-like posets.

Abstract

We consider the problem of determining the maximum number of pairs $F\subseteq F'$ in a family $\mathcal{F}\subseteq 2^{[n]}$ that avoids certain posets $P$ of height 2. We show that for any such $P$ the number of pairs is $O(n\binom{n}{\lfloor n/2\rfloor})$ and we find the exact value for the butterfly poset and the $N$ poset. Also, we determine the asymptotics of the maximum number of pairs in containment for some posets of which the Hasse diagram is a path.