On some monotone path problems in line arrangements

Consider a set of lines in the plane. The lines of induce a cell complex, ! " $# , called the arrangement of , whose vertices are the intersection points of the lines, whose edges are the maximal portions of lines containing no vertices and whose two-dimensional cells are the connected components of %'&)('*,+.-0/ 1 . It is assumed that none of the lines is parallel to the 2 -axis. One of the properties of a line arrangement with lines is the maximum possible length, denoted by 354 , of an 6 monotone polygonal line (path) composed of edges of the arrangement. The length is defined as the number of turns of the polygonal line plus one (i.e., the number of segments of the polygonal path). The problem to estimate 3 4 was posed in [4]. The best known lower bound, due to Balogh et. al. [1], is subquadratic: 78 " '&:90;=< > ?.@ 4