, where denotes the inverse Ackermann function, as may be shown using Davenport–Schinzel sequences. The sum of squares of cell complexities in an arrangement is , as can be shown by summing the zones of all lines. The -level of an arrangement is the polygonal chain formed by the edges that have exactly other lines directly below them. The -level is the portion of the arrangement below the -level. Finding matching upper and lower bounds for the complexity of a -level remains a major open problem in discrete geometry;. The best upper bound is , while the best lower bound is . In contrast, the maximum complexity of the -level is known to be . A -level is a special case of a monotone path in an arrangement; that is, a sequence of edges that intersects any vertical line in a single point. However, monotone paths may be much more complicated than -levels: there exist arrangements and monotone paths in these arrangements where the number of points at which the path changes direction is . Although a single cell in an arrangement may be bounded by all lines, it is not possible in general for different cells to all be bounded by lines. Rather, the total complexity of cells is at most , almost the same bound as occurs in the Szemerédi–Trotter theorem on point-line incidences in the plane. A simple proof of this follows from the crossing number inequality: if cells have a total of edges, one can form a graph with nodes (one per cell) and edges (one per pair of consecutive cells on the same line). The edges of this graph can be drawn as curves that do not cross within the cells corresponding to their endpoints, and then follow the lines of the arrangement. Therefore, there are crossings in this drawing. However, by the crossing number inequality, there are crossings. In order to satisfy both bounds, must be . Projective arrangements and projective duality It is often convenient to study line arrangements not in the Euclidean plane but in the projective plane, due to the fact that in projective geometry every pair of lines has a crossing point. In the projective plane, it is not possible to define arrangements using sides of lines, because a line in the projective plane does not separate the plane into two distinct sides. However, one may still define the cells