{
  "id": "1610.07836",
  "version": "v1",
  "published": "2016-10-25T11:34:31.000Z",
  "updated": "2016-10-25T11:34:31.000Z",
  "title": "Classification of crescent configurations",
  "authors": [
    "Rebecca F. Durst",
    "Max Hlavacek",
    "Chi Huynh",
    "Steven J. Miller",
    "Eyvindur A. Palsson"
  ],
  "comment": "23 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $n$ points be in crescent configurations in $\\mathbb{R}^d$ if they lie in general position in $\\mathbb{R}^d$ and determine $n-1$ distinct distances, such that for every $1 \\leq i \\leq n-1$ there is a distance that occurs exactly $i$ times. Since Erd\\H{o}s' conjecture in 1989 on the existence of $N$ sufficiently large such that no crescent configurations exist on $N$ or more points, he, Pomerance, and Pal\\'asti have given constructions for $n$ up to $8$ but nothing is yet known for $n \\geq 9$. Most recently, Burt et. al. had proven that a crescent configuration on $n$ points exists in $\\mathbb{R}^{n-2}$ for $n \\geq 3$. In this paper, we study the classification of these configurations on $4$ and $5$ points through graph isomorphism and rigidity. Our techniques, which can be generalized to higher dimensions, offer a new viewpoint on the problem through the lens of distance geometry and provide a systematic way to construct crescent configurations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-25T11:34:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "classification",
      "construct crescent configurations",
      "distinct distances",
      "general position",
      "graph isomorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}