{
  "abstract": "A family of nonempty sets is $r$-cover-free if no member is contained in the union of at most $r$ other members. Let $T(r)$ denote the least $n$ for which there is an $r$-cover-free family on $n$ points with more than $n$ members. In this paper, we prove that $T(r)\\ge(r+1)^2$ for every integer $r\\ge1$, confirming the strong form of the Erdős--Frankl--Füredi cover-free conjecture. In particular, $T(r)=(r+1)^2$ whenever $r+1$ is a prime power.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24442",
  "author_total": 2,
  "authors": [
    {
      "name": "Yanan Hu"
    },
    {
      "name": "Leilei Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7dr8mm0k068tt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24442",
  "fulltext_url": null,
  "id": "7dr8mm0k068tt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24442",
  "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|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223041252",
  "pdf_url": "https://arxiv.org/pdf/2610.24442",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7dr8mm0k068tt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24442",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24442"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A proof of the strong Erdős--Frankl--Füredi cover-free conjecture",
  "title_text": "A proof of the strong Erdős--Frankl--Füredi cover-free conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/7dr8mm0k068tt",
  "venue": "arXiv",
  "versions": []
}