# The Happy Ending problem: forcing convex polygons

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**Problem**

Any set of \(2^{n-2} + 1\) points in the plane in general position contains a convex \(n\)-gon
Let \( g(n)\) be the smallest number such that any set of \( n\) points in general position contains a convex \( n\)-gon. Paul Erdösand George Szekeres proved upper [1] and lower [2] bounds:

The upper bound has been improved several times, but the lower bound has not budged. It is correct for \( n \le 5\), and is conjectured to be exact.

Fan Chung and Ron Graham [1] made the first (albeit minor) improvement to the upper bound, showing \( g(n) \leq \binom{2n-4}{n-2}\).

Next, Kleitman and Pachter [3] improved this to \( g(n) \leq \binom{2n-4}{n-2} + 7 - 2n\).

Then Tóth and Valtr [4] improved this by about a factor of 2, showing \( g(n) \leq \binom{2n-5}{n-2} + 2\), which they later improved slightly [5] to \( g(n) \leq \binom{2n-5}{n-2} + 1\).