Suppose we have a collection of $n$ curves in the plane. How large a
subcollection can we necessarily find such that either every pair of
curves in the subcollection pairwise intersects or every pair of
curves in the subcollection are pairwise disjoint? The qualitative
answer is, ``quite large.'' We will discuss how to employ different
algebraic, combinatorial, geometric, and topological methods to
tackle this problem and several variants of it. Along the way, we
will come across several old and new conjectures in combinatorial
geometry.

The talk is based on joint work with Janos Pach.