On the number of families avoiding a subposet

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.

Theorems & Definitions (26)

• Theorem 1.1: methuku2017forbiddentomon2019forbidden
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5: Theorem 1.3 in methuku2017forbidden
• Theorem 1.6: Theorem 12 in tomon2019forbidden
• Theorem 1.7
• Definition 2.1
• Definition 2.2
• Definition 2.3