1 - A Happy Ending

Summary

In the early 1930s, Hungarian mathematician Esther Klein made a discovery that, despite its apparent simplicity, would kick off two major lines of research in mathematics. Klein observed that every set of five points in the plane has either three points in a line or four points in a convex quadrilateral. This became one of the first results in the two fields of discrete geometry (the study of combinatorial properties of geometric objects such as points in the Euclidean plane, and the subject of this book) and Ramsey theory (the study of the phenomenon that unstructured mathematical systems often contain highly structured subsystems).

Klein's observation can be proven by a simple case analysis that considers how many of the points belong to their convex hull. The convex hull is a convex polygon, having some of the given points as its vertices and containing the others. It can be defined mathematically in many ways, for instance as the smallest-area convex polygon that contains all of the given points or as the largest-area simple polygon whose vertices all belong to the given points. The convex hull of points that are not all on a line always has at least three vertices (for otherwise it could not enclose a nonzero area) and, for five given points, at most five vertices. If it has five vertices, any four of them forma convex quadrilateral, and if it has four vertices then it is a convex quadrilateral. The remaining possibility for the convex hull is a triangle, with the other two points either part of a line of three points or inside the triangle.When both points are inside, and the line through themmisses the triangle vertices, it alsomisses one side of the triangle. In this case the two interior points and the two points on the missed side forma convex quadrilateral (Figure 1.1).

The challenge of extending and generalizing this observation was taken up by two of Klein's friends, Paul Erdʺos and George Szekeres. They proved that, for every k, a convex k-gon can be found in all large enough sets of points, as long as no three of the points lie on a line. Klein later married Szekeres, and their marriage is commemorated in the name of Erdʺos and Szekeres's result: the happy ending theorem.