{
  "id": "2211.11341",
  "version": "v1",
  "published": "2022-11-21T10:47:19.000Z",
  "updated": "2022-11-21T10:47:19.000Z",
  "title": "An improved threshold for the number of distinct intersections of intersecting families",
  "authors": [
    "Jagannath Bhanja",
    "Sayan Goswami"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A family $\\mathcal{F}$ of subsets of $\\{1,2,\\ldots,n\\}$ is called a $t$-intersecting family if $|F\\cap G| \\geq t$ for any two members $F, G \\in \\mathcal{F}$ and for some positive integer $t$. If $t=1$, then we call the family $\\mathcal{F}$ to be intersecting. Define the set $\\mathcal{I}(\\mathcal{F}) = \\{F\\cap G: F, G \\in \\mathcal{F} \\text{ and } F \\neq G\\}$ to be the collection of all distinct intersections of $\\mathcal{F}$. Frankl proved an upper bound for the size of $\\mathcal{I}(\\mathcal{F})$ of intersecting families $\\mathcal{F}$ of $k$-subsets of $\\{1,2,\\ldots,n\\}$. Frankl's theorem holds for integers $n \\geq 50 k^2$. In this article, we prove an upper bound for the size of $\\mathcal{I}(\\mathcal{F})$ of $t$-intersecting families $\\mathcal{F}$, provided that $n$ exceeds a certain number $f(k,t)$. Along the way, we also improve Frankl's threshold $k^2$ to $k^{3/2+o(1)}$ for the intersecting families.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-21T10:47:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "intersecting family",
      "distinct intersections",
      "upper bound",
      "frankls theorem holds",
      "frankls threshold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}