{
  "id": "1906.11926",
  "version": "v1",
  "published": "2019-06-27T19:28:17.000Z",
  "updated": "2019-06-27T19:28:17.000Z",
  "title": "On existence of integral point sets and their diameter bounds",
  "authors": [
    "Nikolai Avdeev"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A point set $M$ in $m$-dimensional Euclidean space is called an integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on an $(m-1)$-dimensional hyperplane. We improve the linear lower bound for diameter of planar integral point sets. This improvement takes into account some results related to the Point Packing in a Square problem. Then for arbitrary integers $m \\geq 2$, $n \\geq m+1$, $d \\geq 1$ we give a construction of an integral point set $M$ of $n$ points in $m$-dimensional Euclidean space, where $M$ contains points $M_1$ and $M_2$ such that distance between $M_1$ and $M_2$ is exactly $d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-27T19:28:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C10"
    ],
    "keywords": [
      "diameter bounds",
      "dimensional euclidean space",
      "planar integral point sets",
      "linear lower bound",
      "dimensional hyperplane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}