Erdős Problem #654

The problem

Let $f(n)$ be such that, given any $x_1,\ldots,x_n\in \mathbb{R}^2$ with no four points on a circle, there exists some $x_i$ with at least $f(n)$ many distinct distances to other $x_j$. Estimate $f(n)$ - in particular, is it true that

$$f(n)>(1-o(1))n?$$

Or at least

$$f(n) > (1/3+c)n$$

for some $c>0$, for all large $n$?

Statement quoted from erdosproblems.com/654, maintained by Thomas Bloom. Original source: [Er87b,p.168][ErPa90,p.267][Er97e,p.530]. Current status there: open.

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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