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arXiv · 2505.02229

Incidences, tilings, and fields

P. Pylyavskyy·M. Skopenkov
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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.

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P. Pylyavskyy, M. Skopenkov. 2026-03-27. Incidences, tilings, and fields. https://doi.org/10.1093/imrn%2Frnag046

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