{
  "id": "1304.1861",
  "version": "v1",
  "published": "2013-04-06T06:44:15.000Z",
  "updated": "2013-04-06T06:44:15.000Z",
  "title": "A Common Generalization of the Theorems of Erdős-Ko-Rado and Hilton-Milner",
  "authors": [
    "Wei-Tian Li",
    "Bor-Liang Chen",
    "Kuo-Ching Huang",
    "Ko-Wei Lih"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $m$, $n$, and $k$ be integers satisfying $0 < k \\leq n < 2k \\leq m$. A family of sets $\\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\\binom{[n]}{k} \\subseteq \\mathcal{F} \\subseteq \\binom{[m]}{k}$ and any pair of members of $\\mathcal{F}$ have nonempty intersection. Maximum $(m,k,k)$- and $(m,k+1,k)$-intersecting families are determined by the theorems of Erd\\H{o}s-Ko-Rado and Hilton-Milner, respectively. We determine the maximum families for the cases $n = 2k-1, 2k-2, 2k-3$, and $m$ sufficiently large.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-06T06:44:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "common generalization",
      "hilton-milner",
      "erdős-ko-rado",
      "maximum families",
      "nonempty intersection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.1861L"
    }
  }
}