The number of tangencies between two families of curves

2021/11/16 by Balázs Keszegh, Keszegh, Balázs, Dömötör Pálvölgyi +1
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2111.08787

openalex publication_date 2021/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the number of tangencies between the members of two families, each of which consists of n pairwise disjoint curves, can be as large as Ω(n^4/3). We show that from a conjecture about forbidden 0-1 matrices it would follow that this bound is sharp for doubly-grounded families. We also show that if the curves are required to be x-monotone, then the maximum number of tangencies is Θ(nlog n), which improves a result by Pach, Suk, and Treml. Finally, we also improve the best known bound on the number of tangencies between the members of a family of at most t-intersecting curves.

Related

• On the number of tangencies among 1-intersecting x-monotone curves
• On tangencies among planar curves with an application to coloring L-shapes
• Curves on the torus with few intersections
• On the number of touching pairs in a set of planar curves
• Curves on the torus intersecting at most k times