On Triangles in Colored Pseudoline Arrangements
Yan Alves Radtke, Bal\'azs Keszegh, Robert Lauff

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 . We also prove that if we only consider the triangular faces then…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Combinatorial Mathematics · Limits and Structures in Graph Theory · graph theory and CDMA systems
