Engel's Interval Packing Problem in the Boolean Lattice

Abstract

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 $ν_{n;\ell,u}$, 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.