A Helly Type Conjecture

A family of sets is Π ⁿ , or n -pierceable, if there exists a set of n points such that each member of the family contains at least one of them. It is Π _k ⁿ if every subfamily of size k or less is Π ⁿ . Helly's theorem is one of the fundamental results in Combinatorial Geometry. It asserts, in the special case of finite families of convex sets in the plane, that Π ₃ ¹ implies Π ¹ . However, there is no k such that Π _k ² implies 2 -pierceability for all finite families of convex sets in the plane. It is therefore natural to propose the following: Conjecture. There exists a k ₀ such that, for all planar finite families of convex sets , Π _k0 ² implies Π ³ . Proofs of this conjecture for restricted families of convex sets are discussed. Received October 8, 1996, and in revised form August 12, 1997.