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Claim. There is an infinite set $A \subset \mathbb{R}^2$ such that every $n$-point subset of $A$ contains at least $n/2$ points with no three on a line, yet $A$ is not the union of finitely many sets with no three on a line. The question of Problem 846 therefore has the answer no, with $\epsilon = 1/2$ (Theorem 1.1). The construction takes an algebraically independent sequence $(t_i)$ and the points $P_{i,j} = (t_i + t_j,\, t_i^2 + t_i t_j + t_j^2)$ for $i < j$. Three of these points are collinear exactly when their index pairs form a triangle $\{i,j\},\{j,k\},\{i,k\}$ (Claim 2.1), so a subset with no three collinear is a triangle-free subgraph of the complete graph on $\mathbb{N}$. A graph with $n$ edges has a bipartite, hence triangle-free, subgraph with at least $n/2$ edges, which gives the density property; a partition of $A$ into finitely many collinear-free pieces would be a finite coloring of the edges of the infinite complete graph with no monochromatic triangle, which Ramsey's theorem forbids.

Source. M. Putterman, M. Sawhney and G. Valiant, On infinite sets with no 3 on a line, arXiv:2602.21275, posted 2026-02-24 (two pages); digest on the source card. The paper states that the construction and proof were generated by an internal model at OpenAI and written up by the three authors, who are the claimants. The same text was announced on the site's forum on 2026-02-25 with a copy hosted by OpenAI. The paper reports (p. 1, after Theorem 1.1) that Rödl, shown the proof, noted that the theorem also follows from Theorem 1.7 of Reiher, Rödl and Sales (card) after a generic projection, since the collinear triples of $[3]^n$ are its three-term arithmetic progressions; the forum announcement repeats the remark. The remark has no page of its own: it is a derivation reported in this paper, not a manuscript of Rödl's.

Acceptance. The site's curator, T. F. Bloom, marks the problem disproved and credits this paper, beside the independent proof of a DeepMind prover agent with essentially the same construction, on the problem's page at erdosproblems.com (page last edited 2026-04-10, read 2026-10-07); that credit is the reviewed evidence. No journal publication was found on 2026-10-07, so the claim is not refereed, and this paper has no Lean of its own.