Happy Ending Problem

The happy ending problem, named by mathematician Paul Erdős, states that any set of five points in general position in a plane contains a subset of four points that form the vertices of a convex quadrilateral. This theorem is foundational in the field of Ramsey theory. The Erdős–Szekeres conjecture extends this idea, proposing a relationship between the number of points in a set and the largest subset that can form a convex polygon. Specifically, it suggests that the minimum number of points required to guarantee a convex subset of n points is 2n - 2 + 1. While this conjecture remains unproven, various bounds have been established. Erdős and Szekeres also demonstrated that for any positive integer N, sufficiently large sets of points will contain a subset of N points forming a convex polygon. The values of f(N), which denote the minimum number of points needed to ensure a convex N-gon, are known for small N but remain unknown for larger values. Additionally, the existence of empty convex polygons within point sets has been explored, with recent findings confirming that every set of 30 points in general position contains an empty convex hexagon. This area of study continues to evolve, with ongoing research addressing both the existence of convex shapes and the conditions under which they can be formed.

- The happy ending problem asserts that five points in general position yield a convex quadrilateral.

- The Erdős–Szekeres conjecture relates the number of points to the largest convex subset.

- Known values of f(N) exist for small N, but larger values remain unproven.

- Recent research confirms that 30 points in general position contain an empty convex hexagon.

- The study of convex shapes in point sets is an active area of mathematical research.