Erdős Problem #107 — The Erdős–Szekeres Convex Polygon Function

Let $f(n)$ be the least integer $N$ such that every set of $N$ points in $\mathbb{R}^2$, with no three points collinear, contains $n$ points that are the vertices of a convex $n$-gon. Prove that, for every integer $n\ge 3$,

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