Empty Triangles in Complete Topological Graphs
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 be a complete simple topological graph on vertices. The three edges induced by any triplet of vertices in 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 has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least . We settle Harborth's conjecture in the affirmative.