{
  "id": "2205.05394",
  "version": "v1",
  "published": "2022-05-11T10:38:07.000Z",
  "updated": "2022-05-11T10:38:07.000Z",
  "title": "Stability of intersecting families",
  "authors": [
    "Yang Huang",
    "Yuejian Peng"
  ],
  "comment": "28 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "The celebrated Erd\\H{o}s-Ko-Rado theorem \\cite{EKR1961} states that the maximum intersecting $k$-uniform family on $[n]$ is a full star if $n\\ge 2k+1$. Furthermore, Hilton-Milner \\cite{HM1967} showed that if an intersecting $k$-uniform family on $[n]$ is not a subfamily of a full star, then its maximum size achieves only on a family isomorphic to $HM(n,k):= \\Bigl\\{G\\in {[n] \\choose k}: 1\\in G, G\\cap [2,k+1] \\neq \\emptyset \\Bigr\\} \\cup \\Bigl\\{ [2,k+1] \\Bigr\\} $ if $n>2k$ and $k\\ge 4$, and there is one more possibility in the case of $k=3$. Han and Kohayakawa \\cite{HK2017} determined the maximum intersecting $k$-uniform family on $[n]$ which is neither a subfamily of a full star nor a subfamily of the extremal family in Hilton-Milner theorm, and they asked what is the next maximum intersecting $k$-uniform family on $[n]$. Kostochka and Mubayi \\cite{KM2016} gave the answer for large enough $n$. In this paper, we are going to get rid of the requirement that $n$ is large enough in the result by Kostochka and Mubayi \\cite{KM2016} and answer the question of Han and Kohayakawa \\cite{HK2017}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-11T10:38:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "05C65",
      "05D15"
    ],
    "keywords": [
      "intersecting families",
      "full star",
      "uniform family",
      "maximum intersecting",
      "maximum size achieves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}