Cutting lemma and Zarankiewicz's problem in distal structures

Artem Chernikov; David Galvin; Sergei Starchenko

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arXiv · 1612.00908

Cutting lemma and Zarankiewicz's problem in distal structures

Artem Chernikov·David Galvin·Sergei Starchenko

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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"] on the semialgebraic planar Zarankiewicz problem to arbitrary $o$-minimal structures, in particular obtaining an $o$-minimal generalization of the Szemer\'edi-Trotter theorem.

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Artem Chernikov, David Galvin, Sergei Starchenko. 2016-12-03. Cutting lemma and Zarankiewicz's problem in distal structures. https://arxiv.org/abs/1612.00908

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