A counterexample to a conjecture of Grunbaum on piercing convex sets in the plane

A collection of sets F has the (p, q)-property if out of every p elements of F there are q that have a point in common. A transversal of a collection of sets F is a set A that intersects every member of F. Grunbaum conjectured that every family f 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 F contains two disjoint compacta then there is a transversal of cardinality at most 13. (C) 2013 Elsevier B.V. All rights reserved.