Levi's Lemma, pseudolinear drawings of K n , and empty triangles

There are three main thrusts to this article: a new proof of Levi's Enlargement Lemma for pseudoline arrangements in the real projective plane; a new characterization of pseudolinear drawings of the complete graph; and proofs that pseudolinear and convex drawings of K n haven 2 +O ( n log n ) and O( n 2 ), respectively, empty triangles. All the arguments are elementary, algorithmic, and self‐contained.