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 $o$-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 Szemerédi-Trotter theorem.

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