Erdős Problem #960 — Let $r,k\geq 2$ be fixed.

The problem asks whether sufficiently few ordinary lines in a planar point set with no $k$ collinear points force an $r$-point subset whose all connecting lines are ordinary. The proposed negative answer says even quadratically many ordinary lines need not force such a subset.

Known results

• Turán's theorem gives the general upper bound $f_{r,k}(n)\le \left(1-\frac{1}{r-1}\right)\frac{n^2}{2}+1$.

April 2026 claimed disproof

A manuscript claims that for $r\ge 3$, $k\ge 4$, and $n\ge 72$, an elliptic-curve construction has no four collinear points, at least $\frac{n^2}{12}-\frac{10}{3}n$ ordinary lines, and a bipartite ordinary-line graph, hence no required $K_r$. It attributes the proof to an internal OpenAI model; the claim remains unverified.

Current status (as of August 2026): A manuscript claims a complete negative resolution for all $r\ge 3$ and $k\ge 4$, but independent verification is not recorded, so the problem is not settled.