{
  "id": "2001.01910",
  "version": "v1",
  "published": "2020-01-07T06:43:33.000Z",
  "updated": "2020-01-07T06:43:33.000Z",
  "title": "On Cross-intersecting Sperner Families",
  "authors": [
    "W. H. W. Wong",
    "E. G. Tay"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Two sets $\\mathscr{A}$ and $\\mathscr{B}$ are said to be cross-intersecting if $X\\cap Y\\neq\\emptyset$ for all $X\\in\\mathscr{A}$ and $Y\\in\\mathscr{B}$. Given two cross-intersecting Sperner families (or antichains) $\\mathscr{A}$ and $\\mathscr{B}$ of $\\mathbb{N}_n$, we prove that $|\\mathscr{A}|+|\\mathscr{B}|\\le 2{{n}\\choose{\\lceil{n/2}\\rceil}}$ if $n$ is odd, and $|\\mathscr{A}|+|\\mathscr{B}|\\le {{n}\\choose{n/2}}+{{n}\\choose{(n/2)+1}}$ if $n$ is even. Furthermore, all extremal and almost-extremal families for $\\mathscr{A}$ and $\\mathscr{B}$ are determined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-07T06:43:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05",
      "F.2.2"
    ],
    "keywords": [
      "cross-intersecting sperner families",
      "almost-extremal families",
      "antichains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}