Work

Incidences, Tilings, and Fields

Year: 2026

Type: article

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 ${\mathbb{C}}$ and ${\mathbb{R}}$ 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.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnag046

Authors: P Pylyavskyy, M Skopenkov

Institutions: University of Minnesota, King Abdullah University of Science and Technology

Language: English

Topic: Advanced Combinatorial Mathematics

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,534

Funder: King Abdullah University of Science and Technology