{
  "id": "2409.01343",
  "version": "v1",
  "published": "2024-09-02T15:55:51.000Z",
  "updated": "2024-09-02T15:55:51.000Z",
  "title": "Planar point sets with forbidden $4$-point patterns and few distinct distances",
  "authors": [
    "Terence Tao"
  ],
  "comment": "7 pages, no figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that for any large $n$, there exists a set of $n$ points in the plane with $O(n^2/\\sqrt{\\log n})$ distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a question of Erd\\H{o}s. The proof combines an analysis by Dumitrescu of forbidden four-point patterns with an algebraic construction of Thiele and Dumitrescu (to eliminate parallelograms), as well as a randomized transformation of that construction (to eliminate most other forbidden patterns).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-02T15:55:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C10",
      "05B25"
    ],
    "keywords": [
      "planar point sets",
      "distinct distances",
      "forbidden four-point patterns",
      "set determine",
      "algebraic construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}