An extended lower bound on the number of (≤ k)-edges to generalized configurations of points and the pseudolinear crossing number of K_n

Recently, Aichholzer, García, Orden, and Ramos derived a remarkably improved lower bound for the number of (≤ k)-edges in an n-point set, and as an immediate corollary, an improved lower bound on the rectilinear crossing number of K_n. We use simple allowable sequences to extend all their results to the more general setting of simple generalized configurations of points and slightly improve the lower bound on Sylvester's constant from 0.37963 to 0.379688. In other words, we prove that the pseudolinear (and consequently the rectilinear) crossing number of K_n is at least 0.379688 ((n; 4)) + Θ (n³). We use this to determine the exact pseudolinear crossing numbers of K_n and the maximum number of halving pseudolines in an n-point set for n = 10, 11, 12, 13, 15, 17, 19, and 21. All these values coincide with the corresponding rectilinear numbers obtained by Aichholzer et al.