Ramsey-type constructions for arrangements of segments

Improving a result of Karolyi, Pach and Toth, we construct an arrangement of n segments in the plane with at most n^l^o^g^8^/^l^o^g^1^6^9 pairwise crossing or pairwise disjoint segments. We use the recursive method based on flattenable arrangements which was established by Larman, Matousek, Pach and Torocsik.