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

A generalized configuration is a set of n points and ( n 2 ) 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 0.37968 ( n 4 ) + Θ ( n 3 ) to 0.379972 ( n 4 ) + Θ ( n 3 ) .