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Cosmin Pohoata and Dmitrii Zakharov, Convex polytopes from fewer points, arXiv:2208.04878 (posted 2022-08-09); Duke Math. J. 174 (2025), no. 3, 449–471, DOI 10.1215/00127094-2024-0034. The source card is pohoata_2022_convex_polytopes_fewer_points.

The result. Write $f_k(n)$ for the least $N$ such that every $N$ points in general position in $\mathbb{R}^k$ contain $n$ points in convex position. Theorem 1.1 of the paper states that for every $\varepsilon>0$ and every sufficiently large $n$, every set of at least $2^{\varepsilon n}$ points in general position in $\mathbb{R}^3$ contains $n$ points in convex position; that is, $f_3(n)\le 2^{o(n)}$. A generic projection of a general-position set in $\mathbb{R}^k$ to a hyperplane keeps it in general position, and a subset whose projection is in convex position is itself in convex position, so a convex subset found in the projection lifts back to one in the original set (Valtr's argument, as the paper states it on p. 2); hence $f_k(n)\le f_{k-1}(n)$ for $k\ge3$ (the chain $f_2(n)>f_3(n)>\cdots$ that the site's remark notes), and the bound $f_k(n)\le 2^{o(n)}$ follows for every $k\ge3$.

Why this answers the question. The question asks for a constant $c_k>0$ with $f_k(n)>(1+c_k)^n$. A bound $f_k(n)\le 2^{o(n)}$ means that for every $\varepsilon>0$ and all large $n$, $f_k(n)\le 2^{\varepsilon n}$; a constant $c_k>0$ would force $1+c_k\le 2^{\varepsilon}$ for every $\varepsilon>0$, which is impossible. So no such constant exists for any $k\ge3$, and the answer is no. The planar case $k=2$ is different: the Erdős–Szekeres construction gives $f_2(n)\ge 2^{n-2}+1$, and the exact value is Problem 107. The paper also refutes the prediction of Morris and Soltan that $f_k(n)$ grows like $2^{2n/k}$, and its Theorems 1.2 and 1.3 give positive-fraction versions in dimension three and above; neither is needed for this problem.

Acceptance. The paper is refereed: it appeared in Duke Mathematical Journal, volume 174 (2025), issue 3, pages 449–471. The site's curator, Thomas Bloom, labels the problem DISPROVED (site export of 2026-09-04) and credits the result to Pohoata and Zakharov. The site's thread records a short exchange of December 2025 in which a reader asked why a subexponential bound rules out every positive $c_k$ and received the argument given above. This corpus has checked the statement of Theorem 1.1 against the question as recorded on the source card; it has not reviewed the proof.

Formalizations. Two third-party Lean developments formalize the result, both linked above and neither built or audited by this corpus, so neither gives formalized evidence. Collin Yuanjie Ren's submission jsp-000527-cyr in the repository CollinYuanjieRen/awards, pinned at its commit of 2026-09-16, proves Theorem 1.1 and the subexponential bound in every dimension $k\ge3$ with no hypotheses (theorem_one_one, erdos_651_subexponential, erdos_651_disproved and their all-dimensions forms), with only the three standard axioms, by its README; the README credits the mathematics to Pohoata and Zakharov and says that its new code was prepared with Claude Code (Claude Fable 5.1 and Claude Opus) assistance. It completes Boris Alexeev's Erdos651 development in plby/lean-proofs (formal authors Codex and GPT-5.6 Sol, informal authors Pohoata and Zakharov, by its header), pinned at its commit of 2026-09-15, which on its own proves only the conditional theorem erdos_651_of_pohoata_zakharov under the hypothesis hPZ, the paper's conclusion, together with the unconditional incompatibility statement not_erdos_651. The community database lists the problem as disproved (Lean), citing Ren's formalization, as of its last update of that field on 2026-09-16.