Empty triangles in complete topological graphs

Abstract

A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once. Let G be a simple complete topological graph. The three edges induced by any triple of vertices in G form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an empty triangle. In 1998, Harborth [Harborth, Heiko, Empty triangles in drawings of the complete graph, Discrete Mathematics, 191 (1998), 109–111; Brass, Peter, William O. J. Moser and János Pach, “Research Problems in Discrete Geometry” Springer, (2005)] proved that G has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least 2n/3. In this note, we verify Harborthʼs conjecture.