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Preprint Number 1130

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1130. Artem Chernikov, David Galvin and Sergei Starchenko
Cutting lemma and Zarankiewicz's problem in distal structures

Submission date: 3 December 2016


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

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Full text arXiv 1612.00908: pdf, ps.

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