The Maximum Piercing Number for some Classes of Convex Sets with the (4, 3)-property

A finite collection C of closed convex sets in R is said to have a (p, q)-property if among any p members of C some q have a non-empty intersection, and |C| ≥ p. A piercing number of C is defined as the minimal number k such that there exists a k-element set which intersects every member of C. We focus on the simplest non-trivial case in R, i.e., p = 4 and q = 3. It is known that the maximum possible piercing number of a finite collection of closed convex sets in the plane with (4, 3)-property is at least 3 and at most 13. We consider the following three special types of collections of closed convex sets: segments in R, unit discs in the plane and positively homothetic triangles in the plane, in each case only those satisfying (4, 3)-property. We prove that the maximum possible piercing number is 2 for the collections of segments and 3 for the collections of the other two types.