Erd6s–Szekeres conjecture for the convex-position function

A finite set of points in the plane is in general position if no three points are collinear, and it is in convex position if its points are the vertices of a convex polygon. Let $ES(n)$ be the smallest integer $N$ such that every set of at least $N$ points in general position contains $n$ points in convex position. Erd6s–Szekeres conjecture. For every integer $n\geq2$,

$$ES(n)=2^{n-2}+1.$$

The Erd6s–Szekeres theorem guarantees that $ES(n)$ is finite, while the source records the upper bound $ES(n)\leq\binom{2n-4}{n-2}+1$. Determining the exact value remains open and is the classical Happy Ending Problem.