The happy ending problem asks, for a given integer $n>2$, to find the smallest number $g(n)$ of points laid on a plane in general position such that a subset of $n$ points can be made the vertices of a convex $n$-agon.

It is obvious that for $n=3$, just three points in general position are sufficient to create a triangle. For $n=4$, Paul Erdős and Esther Klein (later Szekeres) determined that at least five points are necessary, and Kalbfleisch later determined $g(5)=9$. Much later, George Szekeres (posthumously) and Lindsay Peters published a computer proof [6] that $g(6)=17$.

For higher $n$, Erdős and George Szekeres in 1935 gave the upper bound

Later, in 1961 they gave the lower bound $g(n)\geq 1+2^{{n-2}}$.

New ideas for the upper bound were in the air in the late 1990s, with Chung and Graham showing that if $n\geq 4$, then

while Kleitman and Pachter showed that then

And Géza Tóth and Pavel Valtr in 1998 gave the upper bound

which in 2005 they refined to