Four degrees of separation
J\'anos Pach
City College and Courant Institute (New York)
pach@cims.nyu.edu

We present several problems in combinatorial geometry, whose solution is based on the introduction of certain partial ordering or separation relations on the underlying set. In particular, we discuss the following question of J. Urrutia. Given a family of pairwise disjoint compact convex sets on a sheet of glass, is it true that one can always separate from one another a constant fraction of them using edge-to-edge straight-line cuts? We answer this question in the negative, and establish some lower and upper bounds for the number of separable sets.