Erdős Problem #960

The problem

Let $r,k\geq 2$ be fixed. Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no $k$ points on a line. Determine the threshold $f_{r,k}(n)$ such that if there are at least $f_{r,k}(n)$ many ordinary lines (lines containing exactly two points) then there is a set $A'\subseteq A$ of $r$ points such that all $\binom{r}{2}$ many lines determined by $A'$ are ordinary.

Is it true that $f_{r,k}(n)=o(n^2)$, or perhaps even $\ll n$?

Statement quoted from erdosproblems.com/960, maintained by Thomas Bloom. Original source: [Er84,p.102]. Current status there: disproved.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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