Cutting lemma and Zarankiewicz's problem in distal structures
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 -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 -minimal structures, in particular obtaining an -minimal generalization of the Szemer\'edi-Trotter theorem.
Cite
@article{arxiv.1612.00908,
title = {Cutting lemma and Zarankiewicz's problem in distal structures},
author = {Artem Chernikov and David Galvin and Sergei Starchenko},
journal= {arXiv preprint arXiv:1612.00908},
year = {2020}
}
Comments
v.2: 29 pages, 3 figures; minor corrections/clarifications throughout the article; Theorem 5.7 has been generalized to allow distal cell decompositions of arbitrary exponent t, and more details were added in the proof; accepted to Selecta Mathematica