On the number of families avoiding a subposet

TL;DR

This paper establishes an upper bound on the number of induced P-free families in the Boolean lattice, linking it to the maximum size of such families, and provides related supersaturation results.

Contribution

It introduces a bound on the number of induced P-free families in the Boolean lattice based on their maximum size, extending understanding of poset-free families.

Findings

Bound of 2^{O(La^*(n,P))} for the number of induced P-free families

Largest size of induced P-free family is a key parameter

Supersaturation results related to poset-free families

Abstract

In this paper we show that for any poset $P$ that is not an antichain, the number of induced $P$-free families in the Boolean lattice $2^{[n]}$ is at most $ 2^{O(\mathrm{La}^*(n,P))}$, where $\mathrm{La}^*(n,P)$ denotes the the largest size of an induced $P$-free subfamily of $2^{[n]}$. We also obtain related supersaturation results.