2019/12/17 by Adam S. Jobson, André E. Kézdy, Jobson, Adam S. +7
Mathematics · #52C99 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:52C99
paper · pdf · doi:10.48550/arxiv.1912.08080
13 pages
arxiv created 2019/12/17 · arxiv updated 2019/12/18
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.