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Elements represented as intersections of sets — arXiv:2608.04163

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.04163
License (as archived): CC0-1.0
Archived: 2026-10-31T11:45:57.862Z
SHA-256 of the archived text: b73183e544b034de4a0a3423263106f93e3ac08bf21b2761982c22006f763281
Stable URL of this page: https://metraipolis.com/documento/dc_7ff1724923dc

How to cite

Elements represented as intersections of sets — arXiv:2608.04163 — arXiv (en), https://arxiv.org/abs/2608.04163 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_7ff1724923dc (Metraipolis archive dc_7ff1724923dc, sha256 b73183e544b034de4a0a3423263106f93e3ac08bf21b2761982c22006f763281)

BibTeX

@misc{dc7ff1724923dc,
  title = {Elements represented as intersections of sets — arXiv:2608.04163},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.04163},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_7ff1724923dc; sha256 b73183e544b034de4a0a3423263106f93e3ac08bf21b2761982c22006f763281},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.04163. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

For a natural number $n$ let $[n] = \{1,\ldots,n\}$. We say that a family ${\cal{S}}\subseteq 2^{[n]}$ is \emph{representing} if every singleton set of $[n]$ is an intersection of some sets from ${\cal{S}}$. We show that the smallest possible cardinality of a representing set for $[n]$ is the discrete inverse $s(n)$ of the Sperner's function $n\mapsto \binom{n}{\lfloor n/2\rfloor}$, which by Sperner's Theorem is the maximum number of elements in an antichain in $2^{[n]}$ when viewed as subset (or boolean) lattice. Specifically, $s(n)$ is then the smallest positive integer such that $2^{[n]}$ contains an $n$-element antichain. Some generalization, further applications and asymptotics in terms of the second real branch of the Lambert $W$ function are presented.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-06 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2608.04163 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-31T11:35:10.098Z dalla pagina https://arxiv.org/catchup/math.CO/2026-08-06?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).