On Triangles in Colored Pseudoline Arrangements

2026-01-28

AI Summary

This study investigates the existence of bichromatic triangles in bichromatically colored pseudoline arrangements and establishes upper bounds on the independence number of the associated face hypergraphs. By combining tools from combinatorial geometry and hypergraph theory, the authors prove for the first time that every nontrivial bichromatic coloring of such an arrangement necessarily contains either a bichromatic triangle or a bichromatic quadrilateral. Furthermore, they determine the exact upper bound on the maximum independence number of the general face hypergraph to be ⌈2n/3 − 1⌉, while for the subhypergraph induced only by triangular faces, the independence number is shown to be n − Θ(log n). This work reveals a novel connection between structural properties of bichromatic pseudoline arrangements and extremal problems in hypergraph independence, thereby advancing the interplay between discrete geometry and extremal combinatorics.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the faces in pseudoline arrangements in which the pseudolines are colored with two colors. Bj\"orner, Las Vergnas, Sturmfels, White, and Ziegler conjecture the existence of a two-colored triangle in such arrangements. We consider variants of this problem. We show that in any non-trivial two-coloring of a pseudoline arrangement there exists a two-colored triangle or quadrangle. We also investigate the existence of a bichromatic triangle assuming certain structures on the coloring. Previously, several authors investigated the chromatic number and independence number of hypergraphs whose vertices correspond to the pseudolines of an arrangement and the hyperedges correspond to the faces of the arrangement. We show that the maximum of the independence numbers of such hypergraphs is $\lceil \frac{2}{3}n-1\rceil$. We also prove that if we only consider the triangular faces then this maximum becomes $n-\Theta(\log n)$.

Problem

• pseudoline arrangements
• two-colored triangle
• bichromatic triangle
• hypergraph independence number
• combinatorial geometry

Innovation

• pseudoline arrangements
• bichromatic triangle
• hypergraph independence number
• two-coloring
• combinatorial geometry