Petruska's question on planar convex sets

TL;DR

This paper investigates a geometric covering problem involving convex sets in the plane, proving the conjecture for certain cases and providing counterexamples for others, connecting it to classical hypergraph extremal problems.

Contribution

It extends Petruska's question by confirming the conjecture for k=4 and presenting a counterexample for k=5, linking geometric and hypergraph extremal problems.

Findings

Confirmed the conjecture for k=4

Provided a counterexample for k=5

Connected the problem to Erdős's arrow problems

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.