Incidences, tilings, and fields
Combinatorics
2026-03-31 v3 Geometric Topology
Metric Geometry
Abstract
The master theorem, introduced by Richter-Gebert and generalized 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.
Cite
@article{arxiv.2505.02229,
title = {Incidences, tilings, and fields},
author = {P. Pylyavskyy and M. Skopenkov},
journal= {arXiv preprint arXiv:2505.02229},
year = {2026}
}
Comments
33 pages, 15 figures. The exposition has been improved: Example 3.3 and Proposition 3.4 have been added to resolve one of the open problems, and several references have been added. The proof of Remark 2.8 has been corrected