Bounds on piercing and line-piercing numbers in families of convex sets in the plane

A family of sets has the (p, q) property if among any p members of it some q intersect. It is shown that if a finite family of compact convex sets in R2 has the (p + 1, 2) property then it is pierced by 1 p 2 J + 1 lines. A colorful version of this result is proved as well. As a corollary, the following is proved: Let F be a finite family of compact convex sets in the plane with no isolated sets, and let F' be the family of its pairwise intersections. If F has the (p + 1, 2) property and F' has the (r + 1, 2) property, then F is pierced by (Lr2j2+L2rJ)p points when r 2, and by p points otherwise. The proofs use the topological KKM theorem. (c) 2023 Elsevier B.V. All rights reserved.