{
  "id": "0908.4038",
  "version": "v1",
  "published": "2009-08-27T16:15:58.000Z",
  "updated": "2009-08-27T16:15:58.000Z",
  "title": "Non-representability of finite projective planes by convex sets",
  "authors": [
    "Martin Tancer"
  ],
  "comment": "8 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We prove that there is no d such that all finite projective planes can be represented by convex sets in R^d, answering a question of Alon, Kalai, Matousek, and Meshulam. Here, if P is a projective plane with lines l_1,...,l_n, a representation of P by convex sets in R^d is a collection of convex sets C_1,...,C_n in R^d such that C_{i_1},...,C_{i_k} have a common point if and only if the corresponding lines l_{i_1},...,l_{i_k} have a common point in P. The proof combines a positive-fraction selection lemma of Pach with a result of Alon on \"expansion\" of finite projective planes. As a corollary, we show that for every $d$ there are 2-collapsible simplicial complexes that are not d-representable, strengthening a result of Matousek and the author.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-08-27T16:15:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A35"
    ],
    "keywords": [
      "finite projective planes",
      "convex sets",
      "non-representability",
      "common point",
      "positive-fraction selection lemma"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0908.4038T"
    }
  }
}