The ACO Seminar (2022–2023)

A family $F$ of sets has the $(p,q)$ property if among 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 $F$. 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.