Absolute incidence theorems and tilings

TL;DR

The paper introduces the concept of absolute incidence theorems in plane projective geometry, showing they hold over any ring, and connects them to tilings of surfaces through the master theorem, with some exceptions.

Contribution

It formalizes absolute incidence theorems, links them to surface tilings via the master theorem, and provides an example of an incidence theorem that is not absolute.

Findings

Most classical incidence theorems are absolute incidence theorems.

The master theorem derives incidence theorems from surface tilings.

An example of an incidence theorem not fitting the absolute framework is provided.

Abstract

We give a precise definition of incidence theorems in plane projective geometry and introduce the notion of ``absolute incidence theorems,'' which hold over any ring. Fomin and Pylyavskyy describe how to obtain incidence theorems from tilings of an orientable surface; they call this result the ``master theorem''. Instances of the master theorem are always absolute incidence theorems. As most classically known incidence theorems are instances of the master theorem, they are absolute incidence theorems. We give an explicit example of an incidence theorem involving 13 points that is not an absolute incidence theorem, and therefore is not an instance of the master theorem.