forced polygon | happy ending problem | Klein-Erdős-Szekeres problem

The happy end problem, also called the "happy ending problem, " is the problem of determining for n>=3 the smallest number of points g(n) in general position in the plane (i.e., no three of which are collinear), such that every possible arrangement of g(n) points will always contain at least one set of n points that are the vertices of a convex polygon of n sides. The problem was so-named by Erdős when two investigators who first worked on the problem, Ester Klein and George Szekeres, became engaged and subsequently married. Since three noncollinear points always determine a triangle, g(3) = 3.

convex hull | convex polygon