Engel's Interval Packing Problem in the Boolean Lattice

Let ${\mathcal{B}}_n$ be the Boolean lattice of all subsets of $[n]$ and let ${\mathcal{P}}_{n;\mathrm{\ell },u}$ be the subposet of ${\mathcal{B}}_n$ induced by the consecutive levels $\mathrm{\ell },\mathrm{\ell }+1,\dots ,u$. We determine ${\nu }_{n;\mathrm{\ell },u}$, the maximum size of a family of pairwise disjoint maximal intervals in ${\mathcal{P}}_{n;\mathrm{\ell },u}$, whenever $u\le (n+{\mathrm{\ell }}^2)\mathrm{/}(\mathrm{\ell }+1)$. 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.