Problem 651

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claims/: The 1 claim page of Problem 651, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $c_k>0$?

Status. 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 [[problems/discrete_geometry/E0651/claims/2022_08_09_pohoata_zakharov|their 2022 result]].

Source. erdosproblems.com/651, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #651, https://www.erdosproblems.com/651.

References.

• [PoZa22] Pohoata, C. and Zakharov, D., Convex polytopes from fewer points. arXiv:2208.04878 (2022); Duke Math. J. 174 (2025), no. 3, 449-471.

Formalization. Two third-party Lean developments, Ren's unconditional one and Alexeev's conditional one, are linked on the claim page; this corpus has built neither, and no formal-conjectures statement is recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• pohoata_2022_convex_polytopes_fewer_points