Tree posets: Supersaturation, enumeration, and randomness

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 $\mathcal {F}$ is a family in $\mathcal {B}_n$ with $|\mathcal {F}|\ge (q-1+\varepsilon ){n\choose \lfloor n/2\rfloor }$ for some $q\ge k$ , then $\mathcal {F}$ 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 [9] and Boehnlein and Jiang [8] which guaranteed a single such copy in non-induced and induced settings, respectively. • The number of induced P -free families of $\mathcal {B}_n$ is $2^{(k-1+o(1)){n\choose \lfloor n/2\rfloor }}$ , strengthening recent independent work of Balogh, Garcia, and Wigal [1] who obtained the same bounds in the non-induced setting. • The largest induced P -free subset of a p -random subset of $\mathcal {B}_n$ 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 [4] and of Collares and Morris [10] 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 [18] in the case of tree posets.