Ramsey-type constructions for arrangements of segments

摘要

Improving a result of Károlyi, Pach and Tóth, we construct an arrangement of n segments in the plane with at most n ~(log8/log169) pairwise crossing or pairwise disjoint segments. We use the recursive method based on flattenable arrangements which was established by Larman, Matou?ek, Pach and T?rocsik.