Engel's Interval Packing Problem in the Boolean Lattice

Let $\mathcal{B}_n$ be the Boolean lattice of all subsets of $[n]$ and let $\mathcal{P}_{n;\ell,u}$ be the subposet of $\mathcal{B}_n$ induced by the consecutive levels $\ell,\ell+1,\ldots,u$. We determine $\nu_{n;\ell,u}$, the maximum size of a family of pairwise disjoint maximal intervals in $\mathcal P_{n;\ell,u}$, whenever $u\le ({n+\ell^2})/({\ell+1})$. This completely settles Engel's problem~[Combin. Probab. Comput., 1996]. The proof is constructive. We also record consequences for weakly cross-intersecting set-pair systems and discuss the three-level case.