{
  "abstract": "Two integer partitions $t$-intersect if they have at least $t$ common parts, counted with multiplicity. We study the largest $t$-intersecting families of integer partitions of $n$ into exactly $k$ positive parts. The canonical $t$-star consists of the partitions containing at least $t$ ones. Applying Kupavskii's weak-spread theorem, we prove that this star is largest whenever $n\\ge Ak^3$, for every fixed $A>24$ and all sufficiently large $k$, uniformly over $1\\le t<k$. We also give counterexamples to Borg's conjecture at every intersection scale: for all sufficiently large $k$ and for every $1\\le t\\le \\lceil k/4\\rceil$, we describe $t$-intersecting families on integer partitions of $n=\\lfloor tk^2/3\\rfloor$ into $k$ parts which are strictly larger than the corresponding canonical stars. Moreover we present a family of counterexamples to Borg's conjectures about largest $t$-intersecting families of integer partitions of $n$ without restricting the number of parts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01747",
  "author_total": 2,
  "authors": [
    {
      "name": "Yury Person"
    },
    {
      "name": "Thomas Schweser"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7r4fn0h9s9bf5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01747",
  "fulltext_url": null,
  "id": "7r4fn0h9s9bf5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01747",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7215872859",
  "pdf_url": "https://arxiv.org/pdf/2610.01747",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7r4fn0h9s9bf5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.01747",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01747"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Intersecting integer partitions: star bounds and counterexamples at every scale",
  "title_text": "Intersecting integer partitions: star bounds and counterexamples at every scale",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/7r4fn0h9s9bf5",
  "venue": "arXiv",
  "versions": []
}