An o-minimal Szemer\'edi-Trotter theorem
  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. 
Saugata Basu and Orit E. Raz
math.LO cs.CG math.CO