Submission date: 3 December 2016

Abstract:

We establish a cutting lemma for definable families of sets in distal structures, as well as the optimality of the distal cell decomposition for definable families of sets on the plane in o-minimal expansions of fields. Using it, we generalize the results in [J. Fox, J. Pach, A. Sheffer, A. Suk, and J. Zahl. “A semi-algebraic version of Zarankiewicz's problem”, Preprint, arXiv:1407.5705 (2014)] on the semialgebraic planar Zarankiewicz problem to arbitrary o-minimal structures, in particular obtaining an o-minimal generalization of the Szemerédi-Trotter theorem.

Mathematics Subject Classification: 03C45, 03C64, 05C35, 05D40

Keywords and phrases:

Full text arXiv 1612.00908: pdf, ps.