{
  "id": "0806.2023",
  "version": "v1",
  "published": "2008-06-12T09:13:46.000Z",
  "updated": "2008-06-12T09:13:46.000Z",
  "title": "Shadows and intersections: stability and new proofs",
  "authors": [
    "Peter Keevash"
  ],
  "comment": "18 pages",
  "journal": "Adv. Math. 218 (2008), 1685--1703",
  "doi": "10.1016/j.aim.2008.03.023",
  "categories": [
    "math.CO"
  ],
  "abstract": "We give a short new proof of a version of the Kruskal-Katona theorem due to Lov\\'asz. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation of Lov\\'asz's theorem that answers a question of Frankl and Tokushige.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-12T09:13:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D05"
    ],
    "keywords": [
      "stability result",
      "approximate structure",
      "combinatorial proof",
      "kruskal-katona theorem",
      "lovaszs theorem"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Adv. Math."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.2023K"
    }
  }
}