Ramsey-type constructions for arrangements of segments

TL;DR

This paper improves bounds on arrangements of segments in the plane, showing new constructions with specific crossing and disjoint properties, and explores limitations of flattenable arrangements and their intersection graphs.

Contribution

It introduces a new recursive construction for segment arrangements with bounded crossing/disjoint pairs and demonstrates limitations of flattenable arrangements through novel intersection graph examples.

Findings

Constructed arrangements with at most n^{log8 / log169} crossing or disjoint segments.

Showed not all arrangements can be flattened, via specific intersection graph examples.

Identified intersection graphs crossing a common line that are not realizable by flattenable arrangements.

Abstract

Improving a result of K\'arolyi, Pach and T\'oth, 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, Matou\v{s}ek, Pach and T\"or\H{o}csik. We also show that not every arrangement can be flattened, by constructing an intersection graph of segments which cannot be realized by an arrangement of segments crossing a common line. Moreover, we also construct an intersection graph of segments crossing a common line which cannot be realized by a flattenable arrangement.