A simple proof of a $(p,2)$-theorem for non-piercing regions

2025-07-09

AI Summary

This paper addresses the piercing problem for families of non-piercing regions in the plane satisfying the $(p,2)$-property—i.e., among any $p$ regions, some two intersect. The goal is to determine whether a finite point set can pierce the entire family. Prior work achieved an $O(p^9)$ upper bound on the minimum number of required points. Here, under the broader non-piercing region model, we exploit structural properties of the intersection hypergraph and map the problem to a hypergraph defined on a hereditary linear Delaunay graph. Combining this geometric reduction with hypergraph piercing theory, we provide a concise combinatorial proof that the piercing number is bounded by $O(p)$. This constitutes the first linear upper bound for this setting. Our result not only substantially improves the best-known bound but also unifies and simplifies the proof framework, offering novel insights and key technical tools for the general $(p,q)$-piercing problem.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

A family of sets satisfies the $(p,2)$-property if among any $p$ sets in the family, some two intersect. Two recent works used elaborate geometric techniques to show that any family of non-piercing regions in the plane that satisfies the $(p,2)$-property can be pierced by $O(p^9)$ points. In this note we show that even in a much more general setting, piercing by $O(p)$ points can be deduced from known results on hypergraphs with a hereditarily linear Delaunay graph, which include intersection hypergraphs of non-piercing regions.

Problem

• Proves a (p,2)-theorem for non-piercing regions
• Simplifies piercing bound to O(p) points
• Generalizes results using hypergraph theory

Innovation

• Non-piercing regions satisfy $(p,2)$-property
• Piercing by $O(p)$ points achieved
• Uses hereditarily linear Delaunay graphs