happy ending problem

Call a finite set $S \subset \mathbb{R}^2$ in general position if no three of its points are collinear, and say that $n$ points of $S$ are in convex position if they are precisely the vertices of a convex $n$-gon. For an integer $n \ge 3$, let

$$f(n) = \min\{\, N \in \mathbb{N} : \text{every set of } N \text{ points in the plane in general position contains } n \text{ points in convex position} \,\},$$

which is a well-defined positive integer for every $n \ge 3$.

Then for every integer $n \ge 3$,

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