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Updated Oct 8

Problem 91

Statement. Let $n$ be a sufficiently large integer. Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between points in $A$ . Prove that there are at least two (and probably many) such $A$ which are non-similar.

Status. Open.

Source. erdosproblems.com/91, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #91, https://www.erdosproblems.com/91.

References.

[Er87b] Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.
[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.
[Ko24c] Z. Kovács, A note on Erdős's mysterious remark. arXiv:2412.05190 (2024).

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (6)

Kovács's computer-algebra proof of the planar case Distance Problems discrete_geometry/erdos_1987_combinatorial_metric_problems_geometry The uniqueness question, p. 171: can the minimum number of distinct distances among n points always be attained in more than one way for large n?Distance Problems distance_problems/kovacs_2024_note_erdos_s_mysterious_remark

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