{
  "id": "2202.10365",
  "version": "v2",
  "published": "2022-02-21T17:00:44.000Z",
  "updated": "2023-04-24T06:37:35.000Z",
  "title": "A proof of Frankl's conjecture on cross-union families",
  "authors": [
    "Stijn Cambie",
    "Jaehoon Kim",
    "Hong Liu",
    "Tuan Tran"
  ],
  "comment": "14 pages, 1 figure Accepted at Combinatorial Theory",
  "categories": [
    "math.CO"
  ],
  "abstract": "The families $\\mathcal F_0,\\ldots,\\mathcal F_s$ of $k$-element subsets of $[n]:=\\{1,2,\\ldots,n\\}$ are called cross-union if there is no choice of $F_0\\in \\mathcal F_0, \\ldots, F_s\\in \\mathcal F_s$ such that $F_0\\cup\\ldots\\cup F_s=[n]$. A natural generalization of the celebrated Erd\\H{o}s--Ko--Rado theorem, due to Frankl and Tokushige, states that for $n\\le (s+1)k$ the geometric mean of $\\lvert \\mathcal F_i\\rvert$ is at most $\\binom{n-1}{k}$. Frankl conjectured that the same should hold for the arithmetic mean under some mild conditions. We prove Frankl's conjecture in a strong form by showing that the unique (up to isomorphism) maximizer for the arithmetic mean of cross-union families is the natural one $\\mathcal F_0=\\ldots=\\mathcal F_s={[n-1]\\choose k}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-04-24T06:37:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "cross-union families",
      "frankls conjecture",
      "arithmetic mean",
      "natural generalization",
      "geometric mean"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}