{
  "id": "1709.02829",
  "version": "v1",
  "published": "2017-09-08T18:33:18.000Z",
  "updated": "2017-09-08T18:33:18.000Z",
  "title": "Diversity of uniform intersecting families",
  "authors": [
    "Andrey Kupavskii"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A family $\\mathcal f\\subset 2^{[n]}$ is called {\\it intersecting}, if any two of its sets intersect. Given an intersecting family, its {\\it diversity} is the number of sets not passing through the most popular element of the ground set. Frankl made the following conjecture: for $n> 3k>0$ any intersecting family $\\mathcal f\\subset {[n]\\choose k}$ has diversity at most ${n-3\\choose k-2}$. This is tight for the following \"two out of three\" family: $\\{F\\in {[n]\\choose k}: |F\\cap [3]|\\ge 2\\}$. In this note we prove this conjecture for $n\\ge ck$, where $c$ is a constant independent of $n$ and $k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-08T18:33:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "intersecting family",
      "uniform intersecting families",
      "sets intersect",
      "popular element",
      "ground set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}