Levi's Lemma, pseudolinear drawings of $K_n$, and empty triangles

TL;DR

This paper presents new elementary proofs and characterizations related to pseudoline arrangements and drawings of complete graphs, focusing on properties like empty triangles and their bounds.

Contribution

It introduces a new proof of Levi's Lemma, a characterization of pseudolinear graph drawings, and bounds on empty triangles in pseudolinear and convex drawings of $K_n$.

Findings

Elementary proof of Levi's Enlargement Lemma

Characterization of pseudolinear drawings of $K_n$

Bounds of $n^2 + O(n \log n)$ and $O(n^2)$ on empty triangles

Abstract

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 $K_n$ have $n^2+{}$O$(n\log n)$ and O$(n^2)$, respectively, empty triangles. All the arguments are elementary, algorithmic, and self-contained.