{
  "id": "1808.03453",
  "version": "v1",
  "published": "2018-08-10T08:40:14.000Z",
  "updated": "2018-08-10T08:40:14.000Z",
  "title": "Stability for Intersecting Families of Perfect Matchings",
  "authors": [
    "Nathan Lindzey"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A family of perfect matchings of $K_{2n}$ is $intersecting$ if any two of its members have an edge in common. It is known that if $\\mathcal{F}$ is family of intersecting perfect matchings of $K_{2n}$, then $|\\mathcal{F}| \\leq (2n-3)!!$ and if equality holds, then $\\mathcal{F} = \\mathcal{F}_{ij}$ where $ \\mathcal{F}_{ij}$ is the family of all perfect matchings of $K_{2n}$ that contain some fixed edge $ij$. In this note, we show that the extremal families are stable, namely, that for any $\\epsilon \\in (0,1/\\sqrt{e})$ and $n > n(\\epsilon)$, any intersecting family of perfect matchings of size greater than $(1 - 1/\\sqrt{e} + \\epsilon)(2n-3)!!$ is contained in $\\mathcal{F}_{ij}$ for some edge $ij$. The proof uses the Gelfand pair $(S_{2n},S_2 \\wr S_n)$ along with an isoperimetric method of Ellis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-10T08:40:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "intersecting family",
      "isoperimetric method",
      "extremal families",
      "equality holds",
      "gelfand pair"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}