Petruska’s question on planar convex sets
Adam S. Jobson, André E. Kézdy, Jenő Lehel, Timothy J. Pervenecki, Gézá Tóth · Discrete Mathematics · 2020
Given 2k−1 convex sets in R2 such that no point of the plane is covered by more than k of the sets, is it true that there are two among the convex sets whose union contains all k-covered points of the plane? This question due to Gy. Petruska has an obvious affirmative answer for k=1,2,3; we show here that the claim is also true for k=4, and we present a counterexample for k=5. We explain how Petruska’s geometry question fits into the classical hypergraph extremal problems, called arrow problems, proposed by P. Erdős.