Tree Posets: Supersaturation, Enumeration, and Randomness

TL;DR

The paper advances extremal poset theory by establishing a balanced supersaturation framework for tree posets P of height k in the Boolean lattice B_n. It introduces a robust embedding tool based on nested sequences of q marked chains and a rank function to map copies to the middle levels M^*(n,q,P), yielding asymptotically tight lower bounds on induced copies and tight counting bounds for induced P free families. Leveraging balanced supersaturation with the hypergraph container method, it derives sharp enumerative results for induced P free families in B_n and for the largest induced P free subset in p random subfamilies, resolving conjectures of Gerbner, Nagy, Patkós, and Vizer in the tree poset case. The work also furnishes a general toolkit for embeddings in poset lattices, with potential applications to random Turán type problems and induced subposet enumeration.

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, Patkós, and Vizer in the case of tree posets.

Theorems & Definitions (65)

• Conjecture 1.1: BukhBukh, Griggs-Lu GL
• Theorem 1.2: Bukh Bukh
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 4.1: Bukh
• Lemma 4.2
• proof