{
  "id": "2108.07087",
  "version": "v1",
  "published": "2021-08-16T13:40:39.000Z",
  "updated": "2021-08-16T13:40:39.000Z",
  "title": "Fixing a hole",
  "authors": [
    "David Conlon",
    "Jeck Lim"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that any finite $S \\subset \\mathbb{R}^d$ in general position has arbitrarily large supersets $T \\supseteq S$ in general position with the property that $T$ contains no empty convex polygon, or hole, with $C_d$ points, where $C_d$ is an integer that depends only on the dimension $d$. This generalises results of Horton and Valtr which treat the case $S = \\emptyset$. The key step in our proof, which may be of independent interest, is to show that there are arbitrarily small perturbations of the set of lattice points $[n]^d$ with no large holes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-16T13:40:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C07"
    ],
    "keywords": [
      "general position",
      "empty convex polygon",
      "large holes",
      "arbitrarily large supersets",
      "arbitrarily small perturbations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}