{
  "id": "0905.0298",
  "version": "v1",
  "published": "2009-05-04T00:10:22.000Z",
  "updated": "2009-05-04T00:10:22.000Z",
  "title": "Point-sets in general position with many similar copies of a pattern",
  "authors": [
    "Bernardo M. Ábrego",
    "Silvia Fernández-Merchant"
  ],
  "comment": "May 3 version. 21 pages, 10 figures",
  "journal": "Geombinatorics 19 (2010), no. 4, 133-145",
  "categories": [
    "math.CO"
  ],
  "abstract": "For every pattern $P$, consisting of a finite set of points in the plane, $S_{P}(n,m)$ is defined as the largest number of similar copies of $P$ among sets of $n$ points in the plane without $m$ points on a line. A general construction, based on iterated Minkovski sums, is used to obtain new lower bounds for $S_{P}(n,m)$ when $P$ is an arbitrary pattern. Improved bounds are obtained when $P$ is a triangle or a regular polygon with few sides. It is also shown that $S_{P}(n,m)\\geq n^{2-\\epsilon}$ whenever $m(n)\\to \\infty$ as $n \\to\\infty$. Finite sets with no collinear triples and not containing the 4 vertices of any parallelogram are called \\emph{parallelogram-free}. The more restricted function $S_{P} ^{\\nparallel}(n)$, defined as the maximum number of similar copies of $P$ among parallelogram-free sets of $n$ points, is also studied. It is proved that $\\Omega(n\\log n)\\leq S_{P}^{\\nparallel}(n)\\leq O(n^{3/2})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-05-04T00:10:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C10"
    ],
    "keywords": [
      "similar copies",
      "general position",
      "finite set",
      "point-sets",
      "parallelogram-free sets"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.0298A"
    }
  }
}