On Order Types of Systems of Segments in the Plane

摘要

Let r(n) denote the largest integer such that every family C of n pairwise disjoint segments in the plane in general position has r(n) members whose order type can be represented by points. Pach and T6th gave a construction that shows r(n) < n~(log 8/log 9) (Pach and Toth 2009). They also stated that one can apply the Erdos-Szekeres theorem for convex sets in Pach and Toth (Discrete Comput Geom 19:437-445,1998) to obtain r(n) > log_(16) n. In this note, we will show that r(n) > cn~(1/4) for some absolute constant c.