Review of Forbidden Configurations in Discrete Geometry by David Eppstein

Abstract

In 1930, the mathematician Esther Klein observed that any five points in the plane in general position (i.e., no three points forming a line) contain four points forming a convex quadrilateral. This innocentsounding discovery led to major lines of research in discrete geometry. Klein's friends Paul Erdős and George Szekeres generalized this theorem, and also conjectured that 2k-2 + 1 points (again in general position) would be enough to force a convex k-gon to exist. The resolution of this conjecture became known as the "happy ending problem," because Klein and Szekeres ended up getting married. The unhappy side is that it has, to date, not been completely solved, although a recent breakthrough of Suk made significant progress. This both mathematically and personally charming little story is a great beginning for this elegant book about discrete geometry. It typifies the type of problems that are studied throughout, and also captures the spirit of curiosity that drives such studies. The book covers many problems that lie at the intersection of three fields: discrete geometry, algorithms and computational complexity.