{
  "abstract": "Fix positive integers $N,k,n$ with $n\\ge k$. We seek the minimum number of members of size at least $n$ in a finite family of finite sets closed under union and containing exactly $N$ distinct sets of size $k$. This problem is a specialization of the Leck--Roberts--Simpson weighted conjecture: assign weight one to sets of size at least $n$ and zero to smaller sets. The predicted minimizer consists of the unions of nonempty subfamilies of the first $N$ $k$-subsets of the natural numbers, ordered by their largest elements and, when these agree, by their increasing lists lexicographically. For an integer $t\\ge 1$, call the range \\[ \\binom{n+t-1}{k}<N\\le\\binom{n+t}{k} \\] the $t$-th strip. We prove the layered conjecture throughout the first strip, and throughout the second strip for $k=3$. For arbitrary $k$, we prove the second strip for families of subsets of an $(n+2)$-element set. For $k,t\\ge 3$, we prove the $t$-th strip for families of subsets of an $(n+t)$-element set whenever $n\\ge(t+1)(k-1)$. With no restriction on the ground set, we prove it for $k\\ge 3$ and $t\\ge 2$ whenever $n>\\frac{5}{2} k^2t$. For sufficiently large $k$, we obtain a sufficient bound of order $k^2t/\\log k$, uniformly in $t\\ge2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11358",
  "author_total": 1,
  "authors": [
    {
      "name": "Amir Jafari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5m6wg71ad49gt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11358",
  "fulltext_url": null,
  "id": "5m6wg71ad49gt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11358",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212218023",
  "pdf_url": "https://arxiv.org/pdf/2609.11358",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5m6wg71ad49gt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11358",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11358"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On union-closed families with prescribed number of $k$-sets",
  "title_text": "On union-closed families with prescribed number of k-sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/5m6wg71ad49gt",
  "venue": "arXiv",
  "versions": []
}