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 without nlog8/log169 pairwise crossing or pairwise disjoint segments. We use the recursive method based on flattenable arrangements which was established by Larman, Matoušek, Pach and Törőcsik. We also discuss some related problems.