Regular systems of paths and families of convex sets in convex position

Abstract:

In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five is in convex position. (If each pair of bodies has at most two common tangents it is enough to assume that every triple is in convex position, and likewise, if each pair of bodies has at most four common tangents it is enough to assume that every quadruple is in convex position.) This confirms a conjecture of Pach and Tóth and generalizes a theorem of Bisztriczky and Fejes Tóth. Our results on families of convex bodies are consequences of more general Ramsey-type results about the crossing patterns of systems of graphs of continuous functions $f : [0,1]\to \mathbb {R}$. On our way towards proving the Pach-Tóth conjecture we obtain a combinatorial characterization of such systems of graphs in which all subsystems of equal size induce equivalent crossing patterns. These highly organized structures are what we call regular systems of paths, and they are natural generalizations of the notions of cups and caps from the famous theorem of Erdős and Szekeres. The characterization of regular systems is combinatorial and introduces some auxiliary structures which may be of independent interest.