The Happy Ending Problem (odd name, haha!) can be described in the following statement: "Any set of five points in the plane in general position has a subset of four points that form the vertices of a convex quadrilateral." One of the results that led to the development of Ramsey Theory, it can be proven using a simple case analysis. The Erdős–Szekeres Conjecture states a more general relationship between the number of points in a general-position point set and its largest convex polygon. It remains unproven, but less precise bounds are known.
YOUR TASK: The Erdos-Szekeres Conjecture has been validated for all 3 < N < 6, for integer values N. But for all N>6 the theorem remains unproven. Visit http://en.wikipedia.org/wiki/Happy_Ending_problem to see the formulas Erdos & Szerkeres conjectured themselves, and use them to experiment for yourself. Also investigate the possibility of "empty polygons," polygons constructed by the theorem that contain no other input points in the given plane.
Comment your findings below!