We give a new lower bound for the rectilinear crossing number cr(n) of the complete geometric graph K n . We prove that cr(n) ≥and we extend the proof of the result to pseudolinear drawings of K n .

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The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph K n in the plane is at least Z (n) := 1

Around 1958, Hill described how to draw the complete graph K n with Z (n) := 1 4 n 2 n − 1 2 n − 2 2 n − 3 2 K n in terms of its number of (≤ k)-edges to the topological setting. Finally, we give a complete characterization of crossing minimal 2-page book drawings of K n and show that, up to equivalence, they are unique for n even, but that there exist an exponential number of non-homeomorphic such drawings for n odd.

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 .