On Triangles in Colored Pseudoline Arrangements

TL;DR

This paper investigates the existence of two-colored triangles in pseudoline arrangements, proving new bounds and variants, and explores related hypergraph properties such as independence numbers.

Contribution

It establishes the existence of either a two-colored triangle or quadrangle in any non-trivial two-coloring and analyzes hypergraph independence numbers related to pseudoline arrangements.

Findings

Existence of a two-colored triangle or quadrangle in any non-trivial coloring

Maximum independence number of hypergraphs is approximately 2/3 of the pseudolines

Maximum independence number for triangular faces is roughly n minus logarithmic factors

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 $n-\Theta(\log n)$.