{
  "abstract": "Stability theorems on intersecting families are often formulated via the covering number of set families. Let $M(n,k,t)$ denote the maximum size of a $k$-uniform intersecting family $\\mathcal F\\subseteq \\binom{[n]}{k}$ with covering number $t$. It is known that for any fixed $t$ and fixed $k$ sufficiently large relative to $t$, one has $M(n,k,t)=(p_{t-1}(k)+o(1))(k)\\binom{n}{k}$ as $n$ tends to infinity, where $p_{t-1}(k)$ is the maximum number of covers of size $t-1$ that a $k$-uniform intersecting family can have provided its covering number is $t-1$. Frankl, Ota, and Tokushige conjectured that $p_t(k)=k^t-\\binom{t}{2}k^{t-1}+O_t(k^{t-2})$ for fixed $t$ as $k$ tends to infinity and they also showed that $p_t(k)=k^t-b(t)k^{t-1}+O_t(k^{t-2})$, where $b(t)$ is defined as the extremum of the following minimization problem. Suppose a $k$-uniform intersecting family $\\mathcal F$ of covering number $t$ can be partitioned into $\\mathcal F_1,\\dots,\\mathcal F_t$ such that for every $i$ there exists a set $X_i$ with $X_i \\subseteq F_i$ for all $F_i\\in\\mathcal F_i$ and the $X_i$'s are pairwise disjoint and non-empty. Then $b(t)$ is the minimum of $\\sum_{i=1}^t(k-|X_i|)$ over all such families, partitions and $X_i$'s. In accordance with the conjecture on $p_t(k)$, Frankl, Ota, and Tokushige conjectured $b(t)=\\binom{t}{2}$ for all $t\\ge 2$ and proved $\\frac{1}{\\sqrt{2}}\\lfloor \\frac{t-1}{2}\\rfloor^{3/2}\\le b(t)\\le \\binom{t}{2}$. In this note, we present a counterexample to the conjecture showing $b(t)\\le 2\\sqrt2\\,t^{3/2}\\sqrt{\\log t}$ $+O(t^{3/2})$ and improve the lower bound to $\\frac{2}{3^{3/4}}t^{3/2}-O(t)\\le b(t)$. The counterexample was found by ChatGPT 6.1Sol, the lower bound proof is due to the human author.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28029",
  "author_total": 1,
  "authors": [
    {
      "name": "Balázs Patkós"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6z91me785pnj7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28029",
  "fulltext_url": null,
  "id": "6z91me785pnj7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28029",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223119957",
  "pdf_url": "https://arxiv.org/pdf/2610.28029",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6z91me785pnj7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28029",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28029"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Intersecting families with large covering number - On a conjecture of Frankl, Ota, and Tokushige",
  "title_text": "Intersecting families with large covering number - On a conjecture of Frankl, Ota, and Tokushige",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/6z91me785pnj7",
  "venue": "arXiv",
  "versions": []
}