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 $n-Θ(\log n)$.

翻译：我们研究伪直线排列中的面，其中伪直线被两种颜色着色。Björner、Las Vergnas、Sturmfels、White和Ziegler猜想在此类排列中存在双色三角形。我们考虑该问题的若干变体。我们证明，在任何非平凡的伪直线排列双着色中，都存在双色三角形或四边形。我们还假设着色具有特定结构，研究了双色三角形的存在性。先前，多位学者研究了超图的色数与独立数，其顶点对应于排列的伪直线，超边对应于排列的面。我们证明此类超图的独立数最大值为$\lceil \frac{2}{3}n-1\rceil$。我们还证明，若仅考虑三角形面，则该最大值变为$n-Θ(\log n)$。