On the number of tangencies among 1-intersecting x-monotone curves

2023/05/23 by Eyal Ackerman, Ackerman, Eyal, Balázs Keszegh +1
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #FOS: Computer and information sciences #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2305.13807

openalex publication_date 2023/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \cC be a set of curves in the plane such that no three curves in \cC intersect at a single point and every pair of curves in \cC intersect at exactly one point which is either a crossing or a touching point. János Pach conjectured that the number of pairs of curves in \cC that touch each other is O(|\cC|). We prove this conjecture for x-monotone curves.

Related

• On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves
• Beyond the Richter-Thomassen Conjecture
• A Crossing Lemma for Families of Jordan Curves with a Bounded Intersection Number
• The number of tangencies between two families of curves
• A Crossing Lemma for Jordan Curves