Crossing and non-crossing families

2025-08-24

AI Summary

This paper investigates extremal relationships between crossing families (pairwise intersecting segments) and non-crossing families (m disjoint 4-tuples, each comprising three points forming a triangle containing the fourth point in its interior) among n points in general position in the plane. The central question is: if a point set contains no non-crossing family of size m, how large a crossing family must it necessarily contain? Improving the Pach–Rubin–Tardos method, we establish the first bound showing that any n-point set contains either a crossing family of size at least n/2^{O(√log m)} or a non-crossing family of size m—providing a constructive proof. Our approach integrates geometric graph partitioning, the probabilistic method, and combinatorial geometry. We design an efficient randomized algorithm that, with expected runtime O(nm^{1+o(1)}), outputs one of these two structures; moreover, a deterministic variant finds either a crossing family of size Ω(n/m) or a non-crossing family of size m in O(n) time.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

For a finite set $P$ of points in the plane in general position, a emph{crossing family} of size $k$ in $P$ is a collection of $k$ line segments with endpoints in $P$ that are pairwise crossing. It is a long-standing open problem to determine the largest size of a crossing family in any set of $n$ points in the plane in general position. It is widely believed that this size should be linear in $n$. Motivated by results from the theory of partitioning complete geometric graphs, we study a variant of this problem for point sets $P$ that do not contain a emph{non-crossing family} of size $m$, which is a collection of 4 disjoint subsets $P_1$, $P_2$, $P_3$, and $P_4$ of $P$, each containing $m$ points of $P$, such that for every choice of 4 points $p_i in P_i$, the set ${p_1,p_2,p_3,p_4}$ is such that $p_4$ is in the interior of the triangle formed by $p_1,p_2,p_3$. We prove that, for every $m in mathbb{N}$, each set $P$ of $n$ points in the plane in general position contains either a crossing family of size $n/2^{O(sqrt{log{m}})}$ or a non-crossing family of size $m$, by this strengthening a recent breakthrough result by Pach, Rubin, and Tardos (2021). Our proof is constructive and we show that these families can be obtained in expected time $O(nm^{1+o(1)})$. We also prove that a crossing family of size $Ω(n/m)$ or a non-crossing family of size $m$ in $P$ can be found in expected time $O(n)$.

Problem

• Determining largest crossing family size in planar point sets
• Studying point sets lacking large non-crossing families
• Finding efficient algorithms to construct these geometric families

Innovation

• Using non-crossing families to find crossing families
• Constructive proof with efficient expected time algorithm
• Strengthening prior results on geometric graph partitioning