{
  "id": "2409.06115",
  "version": "v1",
  "published": "2024-09-09T23:43:31.000Z",
  "updated": "2024-09-09T23:43:31.000Z",
  "title": "On the structure of extremal point-line arrangements",
  "authors": [
    "Gabriel Currier",
    "Jozsef Solymosi",
    "Hung-Hsun Hans Yu"
  ],
  "comment": "8 pages, comments welcome!",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this note, we show that extremal Szemer\\'{e}di-Trotter configurations are rigid in the following sense: If $P,L$ are sets of points and lines determining at least $C|P|^{2/3}|L|^{2/3}$ incidences, then there exists a collection $P'$ of points of size at most $k = k_0(C)$ such that, heuristically, fixing those points fixes a positive fraction of the arrangement. That is, the incidence structure and a small number of points determine a large part of the arrangement. The key tools we use are the Guth-Katz polynomial partitioning, and also a result of Dvir, Garg, Oliveira and Solymosi that was used to show the rigidity of near-Sylvester-Gallai configurations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-09T23:43:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C30"
    ],
    "keywords": [
      "extremal point-line arrangements",
      "large part",
      "points determine",
      "small number",
      "near-sylvester-gallai configurations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}