The maximum piercing number for some classes of convex sets with the \((4,3)\)-property (Q1010733)

Summary: A finite collection \({\mathcal C}\) of closed convex sets in \({\mathbb R}^d\) is said to have a \((p,q)\)-property if among any \(p\) members of \({\mathcal C}\) some \(q\) have a non-empty intersection, and \(|{\mathcal C}| \geq p\). A piercing number of \({\mathcal C}\) is defined as the minimal number \(k\) such that there exists a \(k\)-element set which intersects every member of \({\mathcal C}\). We focus on the simplest non-trivial case in \({\mathbb R}^2\), 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 \({\mathbb R}^d\), 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.