2011/12/31 by Muller, Tobias
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1201.0254
A collection of sets \Fscr has the (p,q)-property if out of every p elements of \Fscr there are q that have a point in common. A transversal of a collection of sets \Fscr is a set A that intersects every member of \Fscr. Grünbaum conjectured that every family \Fscr of closed, convex sets in the plane with the (4,3)-property and at least two elements that are compact has a transversal of bounded cardinality. Here we construct a counterexample to his conjecture. On the positive side, we also show that if such a collection \Fscr contains two \em disjoint compacta then there is a transveral of cardinality at most 13.