Problem 654

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 $n$ ?

Source: erdosproblems.com/654

Geometry
Distances

Wiki page

Status

Open

No claim settles this problem.

Open, the site's label. The site's commentary credits Aletheia [Fe26] with disproving the strongest form, $f(n)>(1-o(1))n$ , by points on two lines with no four on a circle; the result is on Feng and coauthors' claim page (2026), a claimed partial claim. Whether $f(n)>(1/3+c)n$ for some $c>0$ remains open, as does the version with no three points on a line.

Er87b: Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.
Er97e: Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
ErPa90: Erdős, P. and Pach, J., Variations on the theme of repeated distances. Combinatorica (1990), 261-269.
Fe26: T. Feng et al, Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems. arXiv:2601.22401 (2026).

Problem 654

Status

Proof claims

Discussion