Erdős Problem #651

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 $c_k>0$?

#651: [Er97e]

geometry | convex

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

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