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Updated Oct 8

Problem 654

claims/: The 1 claim page of Problem 654, one per claimant's result; the problem's standing derives from them.

Statement. 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 $n$ ?

Status. 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, 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.

Source. erdosproblems.com/654, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #654, https://www.erdosproblems.com/654.

References.

[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).

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

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Linked library material

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Linked from (10)

Feng and coauthors' two-axis set with few distances from every point problems/distance_problems/E0654/claims Distance Problems discrete_geometry/erdos_1987_combinatorial_metric_problems_geometry Conjecture (3) and question (4), p. 168: some point of n points in general position has more than (1+c)n/3 distinct distances Distance Problems distance_problems/chojecki_2026_erdos_problem_655_natural_repairs_exact Lemma 2.1 (p. 2): under the local condition A_m every point sees at least ceil((n-1)/m) distances distance_problems/feng_2026_semi_autonomous_mathematics_discovery_gemini_case Theorem 3: an axis construction with no four concyclic points and every point below 3n/4 distances

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