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 $c_k>0$ ?

Source: erdosproblems.com/651

Geometry
Convexity

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Status

SolvedDisproved

An accepted solution exists. The statement is false.

The site labels the problem DISPROVED (export of 2026-09-04) and credits Pohoata and Zakharov, whose subexponential bound $f_3(n)\le 2^{o(n)}$ rules out every constant $c_k>0$ for $k\ge3$ ; the community database lists it as disproved (Lean), citing Ren's formalization, as of its last update of that field on 2026-09-16. The accepted claim is their 2022 result.

Problem 651

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Proof claims

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