$$\varepsilon $$ε-Mnets: Hitting Geometric Set Systems with Subsets

The existence of Macbeath regions is a classical theorem in convex geometry  (Macbeath in Ann Math 56:269–293, 1952), with recent applications in discrete and computational geometry. In this paper, we initiate the study of Macbeath regions in a combinatorial setting and establish near-optimal bounds for several basic geometric set systems.

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