Petruska's question on planar convex sets

Abstract

Given $2k-1$ convex sets in $R^2$ 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. Erd\H{o}s.