Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon. Prove that $f(n) = 2^{n-2} + 1$.

The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20], who prove $$f(n) ≤ 2^{n+O(\sqrt{n\log n})}.$$

[HMPT20] Holmsen, Andreas F. and Mojarrad, Hossein Nassajian and Pach, János and Tardos, Gábor, _Two extensions of the Erdős-Szekeres problem_. J. Eur. Math. Soc. (JEMS) (2020), 3981-3995.