DISPROVED
This has been solved in the negative.
Let $f_k(n)$ denote the smallest integer such that any $f_k(n)$ points in general position in $\mathbb{R}^k$ contain $n$ 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\[f_3(n)\leq 2^{o(n)}.\]
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Additional thanks to: Mehtaab Sawhney
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #651, https://www.erdosproblems.com/651, accessed 2026-04-11
