{
  "id": "1309.3625",
  "version": "v2",
  "published": "2013-09-14T04:29:01.000Z",
  "updated": "2014-03-14T07:33:34.000Z",
  "title": "A Lower Bound on the Crossing Number of Uniform Hypergraphs",
  "authors": [
    "Anurag Anshu",
    "Saswata Shannigrahi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we consider the embedding of a complete $d$-uniform geometric hypergraph with $n$ vertices in general position in $\\mathbb{R}^d$, where each hyperedge is represented as a $(d-1)$-simplex, and a pair of hyperedges is defined to cross if they are vertex-disjoint and contains a common point in the relative interior of the simplices corresponding to them. As a corollary of the Van Kampen-Flores Theorem, it can be seen that such a hypergraph contains $\\Omega(\\frac{2^d}{\\sqrt{d}})$ $n\\choose 2d$ crossing pairs of hyperedges. Using Gale Transform and Ham Sandwich Theorem, we improve this lower bound to $\\Omega(\\frac{2^d \\log d}{\\sqrt{d}})$ $n\\choose 2d$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-14T07:33:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower bound",
      "uniform hypergraphs",
      "crossing number",
      "uniform geometric hypergraph",
      "van kampen-flores theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.3625A"
    }
  }
}