{
  "id": "1610.03027",
  "version": "v1",
  "published": "2016-10-10T19:00:05.000Z",
  "updated": "2016-10-10T19:00:05.000Z",
  "title": "On the union of intersecting families",
  "authors": [
    "David Ellis",
    "Noam Lifshitz"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that for any integer $r \\geq 2$, if $X$ is an $n$-element set, and $\\mathcal{F} = \\mathcal{F}_1 \\cup \\mathcal{F}_2 \\cup \\ldots \\cup \\mathcal{F}_r$, where each $\\mathcal{F}_i$ is an intersecting family of $k$-element subsets of $X$, then $|\\mathcal{F}| \\leq {n \\choose k} - {n-r \\choose k}$, provided $n \\geq 2k+Ck^{2/3}$, where $C$ is a constant depending upon $r$ alone. Equality holds if and only if $\\mathcal{F} = \\{S \\subset X:\\ |S|=k,\\ S \\cap R \\neq \\emptyset\\}$ for some $R \\subset X$ with $|R|=r$. In the case $r=2$, this improves a result of Frankl and F\\\"uredi.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-10T19:00:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "intersecting family",
      "element set",
      "element subsets",
      "equality holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}