A central approach to bound the number of crossings in a generalized configuration

A generalized configuration is a set of n points and pseudolines such that each pseudoline passes through exactly two points, two pseudolines intersect exactly once, and no three pseudolines are concurrent. Following the approach of allowable sequences we prove a recursive inequality for the number of (⩽k)-sets for generalized configurations. As a consequence we improve the previously best known lower bound on the pseudolinear and rectilinear crossing numbers from to .