On Triangles in Colored Pseudoline Arrangements

TL;DR

The paper addresses the existence of bichromatic faces in two-coloured pseudoline arrangements and analyzes the independence structure of associated line-face and line-triangle hypergraphs. It develops signotope-based and lens-sweeping techniques to study how colourings force small coloured faces and derives exact and near-exact bounds on independence numbers, notably proving that non-trivial two-colourings yield a bichromatic triangle or quadrangle and giving precise results for psline-face hypergraphs. The work advances understanding of colored planar arrangements and their hypergraph representations, including tight bounds and constructive lower bounds that distinguish pseudolines from straight-line realizations. These results have implications for oriented matroids, face-structure combinatorics, and related geometric hypergraph problems, contributing both exact results and guiding open questions.

Abstract

We consider the faces in pseudoline arrangements in which the pseudolines are colored with two colors. Björner, 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 $n-Θ(\log n)$.

Figures (13)

• Figure 1: The north face is designated by $N$. The face $T$ is a triangle. The crossings (1,2) and (1,3) are vertices, which are connected by an edge. The left arrangement corresponds to the all "$-$" signotope. The right arrangement is obtained by flipping $T$.
• Figure 2: The two cyclic arrangements on 8 pseudolines. Note that these arrangements are realizable by lines.
• Figure 3: The arrangement on the left is block-bicolored. The arrangement on the right is not.
• Figure 4: The 5-star is on the left side. On the right side, there is the situation after flipping the first triangle and finding a new triangle.
• Figure 5: Both extremal arrangements with prescribed sub-arrangements

Theorems & Definitions (22)

• Conjecture 1: orientedmatroids
• Conjecture 2: orientedmatroids
• Definition 1: sweeps
• Lemma 1: sweeps
• Theorem 1: theoremonhbo
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 1
• Theorem 5: firstsecondBS18