A Helly type conjecture (Q1283738)

We say that a family of sets is \(\Pi^n\) provided there exists a set of \(n\) points such that every set from the family contains at least one of those points. A family of sets is said to be \(\Pi^n_k\) if each its subfamily of at most \(k\) sets is \(\Pi^n\). The authors conjecture that there exists a positive integer \(k_0\) such that if a family of convex sets in the plane is \(\Pi^2_{k_0}\), then it is also \(\Pi^3\). They show that if such an integer \(k_0\) exists, then it is over 5. The paper confirms the above conjecture for two very special cases. One of them is when \(k_0= 10\) and when the family of convex sets contains only intersections of translates of any four fixed closed half-planes.