Ramsey-type constructions for arrangements of segments

Improving a result of Károlyi, Pach and Tóth, we construct an arrangement of n segments in the plane with at most n log 8 / log 169 pairwise crossing or pairwise disjoint segments. We use the recursive method based on flattenable arrangements which was established by Larman, Matoušek, Pach and Törőcsik.