A counterexample to conjecture 18.5 in “Geometric Etudes in Combinatorial Mathematics”, second edition

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. Grünbaum 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 transveral of cardinality at most 13. 1 Introduction and statement of results Let F be a collection of sets. A transversal of F is a set A that intersects every member of F (that is, A ∩ F 6 = ∅ for all F ∈ F). The transversal number of piercing number τ(F) of F is the smallest size of a transversal, i.e. τ(F): = mi