On the maximum number of tangencies among $1$-intersecting curves

2026-03-12

AI Summary

This study investigates the maximum number of tangencies among $n$ Jordan arcs in the plane under the condition that any two arcs intersect at most once and no three arcs meet at a common point. By employing tools from combinatorial geometry and extremal graph theory, the authors improve the upper bounds on the number of tangencies to $O(n^{5/3})$ in the general setting and to $O(n^{3/2})$ under the stricter assumption of simple 1-intersecting arcs. For $x$-monotone arcs, they establish a tight bound of $\Theta(n^{4/3})$ and further refine the upper bound to $O(n^{4/3}(\log n)^{1/3})$. Additionally, they prove a graph-theoretic result generalizing the Erdős–Simonovits theorem, which substantially advances the known upper bounds for tangency counts in cases related to Pach’s conjecture.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Applications and emergent phenomena

Abstract

According to a conjecture of Pach, there are $O(n)$ tangent pairs among any family of $n$ Jordan arcs in which every pair of arcs has precisely one common point and no three arcs share a common point. This conjecture was proved for two special cases, however, for the general case the currently best upper bound is only $O(n^{7/4})$. This is also the best known bound on the number of tangencies in the relaxed case where every pair of arcs has \emph{at most} one common point. We improve the bounds for the latter and former cases to $O(n^{5/3})$ and $O(n^{3/2})$, respectively. We also consider a few other variants of these questions, for example, we show that if the arcs are \emph{$x$-monotone}, each pair intersects at most once and their left endpoints lie on a common vertical line, then the maximum number of tangencies is $Θ(n^{4/3})$. Without this last condition the number of tangencies is $O(n^{4/3}(\log n)^{1/3})$, improving a previous bound of Pach and Sharir. Along the way we prove a graph-theoretic theorem which extends a result of Erdős and Simonovits and may be of independent interest.

Problem

• tangencies
• Jordan arcs
• intersection patterns
• combinatorial geometry
• extremal graph theory

Innovation

• tangencies
• Jordan arcs
• combinatorial geometry
• intersection graphs
• extremal graph theory