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 $$G$$G be a complete simple topological graph on $$n$$n vertices. The three edges induced by any triplet of vertices in $$G$$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 proved that $$G$$G has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least $$2n/3$$2n/3. We settle Harborth's conjecture in the affirmative.