Cutting lemma and Zarankiewicz's problem in distal structures

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 ominimal expansions of fields. Using it, we generalize the results in [10] on the semialgebraic planar Zarankiewicz problem to arbitrary o-minimal structures, in particular obtaining an o-minimal generalization of the Szemerédi-Trotter theorem.