Ramsey-type constructions for arrangements of segments

Details Ramsey-type constructions for arrangements of segments Ramsey-type constructions for arrangements of segments

2010-10-11

arxiv.org/abs/1010.2167v1

European Journal of Combinatorics 33 (2012), Issue 3, 336-339

Mathematics

Combinatorics

11 pages, 6 figures

Scientific paper

10.1016/j.ejc.2011.09.006

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.

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