Tree Posets: Supersaturation, Enumeration, and Randomness

TL;DR

This paper introduces a new embedding tool for tree posets in Boolean lattices, solving open problems related to supersaturation, enumeration, and randomness, and confirming conjectures in the area.

Contribution

The authors develop a versatile embedding method for tree posets, leading to tight asymptotic results in supersaturation, enumeration, and random subset analysis.

Findings

Families with large size contain many induced copies of P.

Number of induced P-free families is exponentially small.

Largest induced P-free subset in random sets is tightly bounded.

Abstract

We develop a powerful tool for embedding any tree poset $P$ of height $k$ in the Boolean lattice which allows us to solve several open problems in the area. We show that: * If $H$ is a family in $B_n$ with $|H|\ge (q-1+\varepsilon){n\choose \lfloor n/2\rfloor}$ for some $q\ge k$, then $H$ contains on the order of as many induced copies of $P$ as is contained in the $q$ middle layers of the Boolean lattice. This generalizes results of Bukh and of Boehnlein and Jiang which guaranteed a single such copy in non-induced and induced settings respectively. * The number of induced $P$-free families of $B_n$ is $2^{(k-1+o(1)){n\choose \lfloor n/2\rfloor}}$, strengthening recent independent work of Balogh, Garcia, Wigal who obtained the same bounds in the non-induced setting. * The largest induced $P$-free subset of a $p$-random subset of $B_n$ for $p\gg n^{-1}$ has size at most $(k-1+o(1))p{n\choose \lfloor n/2\rfloor}$, generalizing previous work of Balogh, Mycroft, and Treglown and of Collares and Morris for the case when $P$ is a chain. All three results are asymptotically tight and give affirmative answers to general conjectures of Gerbner, Nagy, Patk\'os, and Vizer in the case of tree posets.