{
  "abstract": "For integers $n\\ge s\\ge2$, let $e(n,s)$ be the maximum size of a family $\\mathcal F\\subseteq2^{[n]}$ with no $s$ pairwise disjoint members. The problem of determining $e(n,s)$, now called the Erdős--Kleitman problem, is closely related to the well-known Erdős matching problem. Frankl and Kupavskii posed a meta-conjecture predicting that the maximum is always attained by a weighted family. Fix $m\\ge3$, write $n=ms+c$ with $0\\le c<s$, and set $\\ell=s-c$. For $0\\le k\\le m$, let $a_k=ms-kc-1$. For $A\\in\\binom{[n]}{a_k}$, define \\[ \\mathcal H^k(m,s,\\ell;A):= \\{F\\subseteq[n]: k|F|+|F\\cap A|\\ge m(k+1)\\}. \\] This defines a unified class of weighted families with matching number less than $s$. Among these families, $\\mathcal H^0$, $\\mathcal H^1$, and $\\mathcal H^m$ were previously known to be extremal in different ranges of $c$. We show that for $1\\le k\\le m-1$, all families $\\mathcal H^k$ are uniquely extremal in some ranges of $c$. More precisely, we prove that for every $m\\ge3$ and every $1\\le k\\le m-1$, there exist constants $α=α(m,k)>0$, $β=β(m,k)>0$ and an integer $s_0=s_0(m,k)$ such that, for all integers $s\\ge s_0$ and all integers $c$ with $0\\le c<s$, the only extremal families for $e(n,s)$ are the families $\\mathcal H^k(m,s,\\ell;A)$ with $A\\in\\binom{[n]}{a_k}$ whenever $βs^{(k-1)/k}\\le c\\le αs^{k/(k+1)}$. In particular, this result determines an infinite number of new extremal families for the Erdős--Kleitman problem and verifies the Frankl--Kupavskii meta-conjecture in these ranges. This also provides a quantitative extension of the result of Kupavskii and Sokolov on the extremality of $\\mathcal H^1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.25611",
  "author_total": 2,
  "authors": [
    {
      "name": "Cheng Chi"
    },
    {
      "name": "Yan Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5pke1e04z43qs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.25611",
  "fulltext_url": null,
  "id": "5pke1e04z43qs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.25611",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7171705105",
  "pdf_url": "https://arxiv.org/pdf/2607.25611",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5pke1e04z43qs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.25611",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.25611"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Extremal Families for the Erdős--Kleitman Problem: The Missing Constructions",
  "title_text": "Extremal Families for the Erdős--Kleitman Problem: The Missing Constructions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/5pke1e04z43qs",
  "venue": "arXiv",
  "versions": []
}