An o-minimal Szemer\'edi-Trotter theorem

TL;DR

This paper extends the Szemerédi-Trotter theorem to definable curves and points within o-minimal structures over real closed fields, introducing new combinatorial tools for such settings.

Contribution

It provides the first o-minimal analog of the Szemerédi-Trotter theorem and develops an o-minimal crossing number inequality for graph embeddings.

Findings

Established an o-minimal Szemerédi-Trotter theorem

Extended crossing number inequality to o-minimal structures

Applicable over arbitrary real closed fields

Abstract

We prove an analog of the Szemer\'edi-Trotter theorem in the plane for definable curves and points in any o-minimal structure over an arbitrary real closed field $\mathrm{R}$. One new ingredient in the proof is an extension of the well known crossing number inequality for graphs to the case of embeddings in any o-minimal structure over an arbitrary real closed field.