Abstract: A family \(F\) of sets has the \((p,q)\) property if amongst any \(p\) members of it some \(q\) intersect. \(F\) has the \(T(k)\) property if every \(k\) sets in \(F\) are intersected by a line. We prove that if \(F\) is a family of convex sets in the plane with the \((p+1,2)\) property then there are \( \lfloor p/2\rfloor +1\) lines whose union intersects all the sets in \(F\), and this bound is tight. We use this result to prove new bounds on the piercing numbers in families of convex sets in the plane with the \((p,2)\) property, in terms of the matching numbers of their pairwise intersection families. We further prove a conjecture of Eckhoff from 1993, asserting that if a family of convex sets in the plane has the \(T(3)\) property then there are 3 lines whose union intersects all the sets in it. Rainbow versions of these results are also proved. The proofs use the topological KKM theorem and its colorful generalization.

Partly joined with Daniel McGinnis.