Containments in families with forbidden subposets

Abstract

We consider the problem of determining the maximum number of pairs $F\subseteq F^{\prime }$ 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 $O(n(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }))$ 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.

merged with arXiv:2108.08898