{
  "id": "2206.09334",
  "version": "v1",
  "published": "2022-06-19T06:57:47.000Z",
  "updated": "2022-06-19T06:57:47.000Z",
  "title": "Maximal 3-wise Intersecting Families with Minimum Size: the Odd Case",
  "authors": [
    "József Balogh",
    "Ce Chen",
    "Haoran Luo"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A family $\\mathcal{F}$ on ground set $\\{1,2,\\ldots, n\\}$ is maximal $k$-wise intersecting if every collection of $k$ sets in $\\mathcal{F}$ has non-empty intersection, and no other set can be added to $\\mathcal{F}$ while maintaining this property. Erd\\H{o}s and Kleitman asked for the minimum size of a maximal $k$-wise intersecting family. Complementing earlier work of Hendrey, Lund, Tompkins and Tran, who answered this question for $k=3$ and large even $n$, we answer it for $k=3$ and large odd $n$. We show that the unique minimum family is obtained by partitioning the ground set into two sets $A$ and $B$ with almost equal sizes and taking the family consisting of all the proper supersets of $A$ and of $B$. A key ingredient of our proof is the stability result by Ellis and Sudakov about the so-called $2$-generator set systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-19T06:57:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimum size",
      "odd case",
      "intersecting family",
      "ground set",
      "generator set systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}