Seminars & Colloquia Calendar

Abstract: Consider a finite family F of geometric objects in the plane (e.g. disks, rectangles, convex sets, strings). Is it possible to colour the members of F with a few colours such that elements of the same colour are pairwise disjoint? Of course, the answer is no if F contains a large subfamily in which any two members have a nonempty intersection. Is this the only obstacle? Now the answer to this question heavily depends on what kind of geometric objects we consider.

Also, how does the problem change if we want a colouring in which any two elements of the same colour intersect? Can we always find a large subset of F where any two elements are disjoint, or any two elements have a nonempty intersection?

In the past 60 years, such problems have been extensively studied both in combinatorial and computational geometry, and are strongly tied to questions in structural graph theory and Ramsey theory. Despite their innocent appearance, these problems are not very well understood even for simple geometric objects, like rectangles. I will present the history of this topic along with recent developments, many of which are due to new results about partial orders and ordered graphs.