{
  "abstract": "A family of subsets of $[n]$ is $r$-fork-free if none of its members is strictly contained in $r$ other distinct members. For each fixed integer $r\\ge2$, we prove that the maximum size of such a family is \\[ \\binom{n}{\\lfloor n/2\\rfloor} \\left(1+\\frac{2(r-1)}{n}+o(n^{-1})\\right). \\] This determines the second-order term and matches the upper bound of De Bonis and Katona. Our lower bound comes from a construction on two adjacent middle levels, using a finite ordered collection of disjoint coordinate blocks and a condition on sums modulo $n$. For each prescribed error, the blocks are fixed before $n$ tends to infinity, so the construction works for every sufficiently large $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30336",
  "author_total": 3,
  "authors": [
    {
      "name": "Yiyan Zhan"
    },
    {
      "name": "Mei Lu"
    },
    {
      "name": "Xiamiao Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/74hp9xhrf71bs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30336",
  "fulltext_url": null,
  "id": "74hp9xhrf71bs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30336",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.30336",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/74hp9xhrf71bs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30336",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30336"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The second-order term for the largest $r$-fork-free families",
  "title_text": "The second-order term for the largest r-fork-free families",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/74hp9xhrf71bs",
  "venue": "arXiv",
  "versions": []
}