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 Szemer\'edi-Trotter theorem.
['Artem Chernikov', 'David Galvin', 'Sergei Starchenko']
Cutting lemma and Zarankiewicz's problem in distal structures