{
  "id": "2304.13466",
  "version": "v1",
  "published": "2023-04-26T11:41:59.000Z",
  "updated": "2023-04-26T11:41:59.000Z",
  "title": "Strong stability of 3-wise $t$-intersecting families",
  "authors": [
    "Norihide Tokushige"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathcal G$ be a family of subsets of an $n$-element set. The family $\\mathcal G$ is called $3$-wise $t$-intersecting if the intersection of any three subsets in $\\mathcal G$ is of size at least $t$. For a real number $p\\in(0,1)$ we define the measure of the family by the sum of $p^{|G|}(1-p)^{n-|G|}$ over all $G\\in\\mathcal G$. For example, if $\\mathcal G$ consists of all subsets containing a fixed $t$-element set, then it is a $3$-wise $t$-intersecting family with the measure $p^t$. For a given $\\delta>0$, by choosing $t$ sufficiently large, the following holds for all $p$ with $0<p\\leq 2/(\\sqrt{4t+9}-1)$. If $\\mathcal G$ is a $3$-wise $t$-intersecting family with the measure at least $(\\frac12+\\delta)p^t$, then $\\mathcal G$ satisfies one of (i) and (ii): (i) every subset in $\\mathcal G$ contains a fixed $t$-element set, (ii) every subset in $\\mathcal G$ contains at least $t+2$ elements from a fixed $(t+3)$-element set.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-26T11:41:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "intersecting family",
      "element set",
      "strong stability",
      "real number",
      "sufficiently large"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}