Long monotone paths in line arrangements (Q1762943)

A path in the arrangement \(A(L)\) (where \(L\) is a set of \(n\) given lines in \(\mathbb{R}^2\)) is a simple polygonal chain joining a set of distinct vertices in \(V\) (where \(V\) is generated by the intersections of the lines in \(L)\) by segments which are on lines in \(L\). The length of a path is the number of vertices in \(V\) at which the path turns plus one. A path is monotone in direction \((a,b)\) if its sequence of vertices is monotone when projected orthogonally onto the line with equation \(ay - bx = 0\). In this paper it is shown how to construct an arrangement of \(n\) lines having a monotone path of length \(\Omega(n^{2-\frac{d}{\sqrt{\log n}}})\), where \(d > 0\) is some constant. At the end of the paper some related questions and the implications of this result are discussed.