Abstract
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 ⌊p2⌋+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 (⌊r2⌋2+⌊r2⌋)p points when r⩾2, and by p points otherwise. The proofs use the topological KKM theorem.
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