{
  "id": "2505.02229",
  "version": "v1",
  "published": "2025-05-04T19:33:09.000Z",
  "updated": "2025-05-04T19:33:09.000Z",
  "title": "Incidences, tilings, and fields",
  "authors": [
    "P. Pylyavskyy",
    "M. Skopenkov"
  ],
  "comment": "31 pages, 14 figures",
  "categories": [
    "math.CO",
    "math.GT",
    "math.MG"
  ],
  "abstract": "The master theorem, introduced independently by Richter-Gebert and by Fomin and the first author, provides a method for proving incidence theorems of projective geometry using triangular tilings of surfaces. We investigate which incidence theorems over C and R can or cannot be proved via the master theorem. For this, we formalize the notion of a tiling proof. We introduce a hierarchy of classes of theorems based on the underlying topological spaces. A key tool is considering the same theorems over finite fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-04T19:33:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51A20",
      "05E14",
      "14N20",
      "51M15",
      "57Q05",
      "12E20"
    ],
    "keywords": [
      "master theorem",
      "first author",
      "finite fields",
      "triangular tilings",
      "proving incidence theorems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}