Erdős Problem #846 — Let $A\subset \mathbb{R}^2$ be an infinite set for which there exists some $\epsilon>0$ such that in any subset of $A$ of size $n$ there are always at least $\epsilon n$ with no three on a line.

Erdős, Nešetřil, and Rödl asked whether an infinite $A\subset\mathbb{R}^2$ in which every sufficiently large finite subset contains linearly many points with no three collinear must be a finite union of such sets.

February–May 2026 counterexample

The explicit construction maps each edge $\{x_i,x_j\}$ of $K_{\infty}$ to $P_e=(x_i+x_j,x_i^2+x_ix_j+x_j^2)$. Collinearity of three points is equivalent to a triangle among the corresponding edges. Every finite edge set has a bipartite subgraph with at least $n/2$ edges, giving $\epsilon=1/2$, while infinite Ramsey theory forbids any finite partition into triangle-free color classes. Independent accounts attribute the construction to a DeepMind prover agent and an internal OpenAI model; arXiv papers provide the mathematical argument.

Current status (as of June 2026): The conjecture is disproved by an explicit construction with $\epsilon=1/2$; nothing remains open in the stated problem.