SOLVED

Let $f_k(n)$ denote the smallest integer such that any $f_k(n)$ points in general position in $\mathbb{R}^k$ contain at $n$ which determine a convex polyhedron. Is it true that
\[f_k(n) > (1+c_k)^n\]
for some constant $c_k>0$?

The function when $k=2$ is the subject of the Erdős-Klein-Szekeres conjecture, see [107]. One can show that
\[f_2(n)>f_3(n)>\cdots.\]
The answer is no, even for $k=3$: Pohoata and Zakharov [PoZa22] have proved that
\[f_3(n)\leq 2^{o(n)}.\]