An o-minimal Szemerédi-Trotter theorem

Abstract

We prove an analog of the Szemerédi-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.

15 pages. Final version to appear in The Quarterly Journal of Mathematics