{
  "id": "2103.15127",
  "version": "v1",
  "published": "2021-03-28T13:28:39.000Z",
  "updated": "2021-03-28T13:28:39.000Z",
  "title": "A stability result on matchings in 3-uniform hypergraphs",
  "authors": [
    "Mingyang Guo",
    "Hongliang Lu",
    "Dingjia Mao"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $n,s,k$ be three positive integers such that $1\\leq s\\leq(n-k+1)/k$ and let $[n]=\\{1,\\ldots,n\\}$. Let $H$ be a $k$-graph with vertex set $\\{1,\\ldots,n\\}$, and let $e(H)$ denote the number of edges of $H$. Let $\\nu(H)$ and $\\tau(H)$ denote the size of a largest matching and the size of a minimum vertex cover in $H$, respectively. Define $A^k_i(n,s):=\\{e\\in\\binom{[n]}{k}:|e\\cap[(s+1)i-1]|\\geq i\\}$ for $2\\leq i\\leq k$ and $HM^k_{n,s}:=\\big\\{e\\in\\binom{[n]}{k}:e\\cap[s-1]\\neq\\emptyset\\big\\} \\cup\\big\\{S\\big\\}\\cup \\big\\{e\\in\\binom{[n]}{k}: s\\in e, e\\cap S\\neq \\emptyset\\}$, where $S=\\{s+1,\\ldots,s+k\\}$. Frankl and Kupavskii conjectured that if $\\nu(H)\\leq s$ and $\\tau(H)>s$, then $e(H)\\leq \\max\\{|A^k_2(n,s)|,\\ldots ,|A^k_k(n,s)|,|HM^k_{n,s}|\\}$. In this paper, we prove this conjecture for $k=3$ and sufficiently large $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-28T13:28:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stability result",
      "hypergraphs",
      "minimum vertex cover",
      "vertex set",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}