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PreprintCC0arXiv

Engel's Interval Packing Problem in the Boolean Lattice

  • Yuxian Dong
  • Jianxi Mao
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Abstract

Let ℬn be the Boolean lattice of all subsets of [n] and let 𝒫n;β„“,u be the subposet of ℬn induced by the consecutive levels β„“,β„“+1,…,u. We determine Ξ½n;β„“,u, the maximum size of a family of pairwise disjoint maximal intervals in 𝒫n;β„“,u, whenever u≀ (n+β„“2)/(β„“+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.

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Cite

BibTeXRIS
BibTeX
@misc{dong2026engels,
  title = {{Engel's Interval Packing Problem in the Boolean Lattice}},
  author = {Yuxian Dong and Jianxi Mao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.04794},
  eprint = {2607.04794},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.04794},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Engel's Interval Packing Problem in the Boolean Lattice
AU  - Yuxian Dong
AU  - Jianxi Mao
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2607.04794
UR  - https://arxiv.org/abs/2607.04794
AB  - Let ℬ_n be the Boolean lattice of all subsets of [n] and let 𝒫_n;β„“,u be the subposet of ℬ_n induced by the consecutive levels β„“,β„“+1,…,u. We determine Ξ½_n;β„“,u, the maximum size of a family of pairwise disjoint maximal intervals in 𝒫_n;β„“,u, whenever u≀ (n+β„“^2)/(β„“+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.
ER  -