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Updated Oct 7

Problem 1087

Statement. Let $f(n)$ be minimal such that every set of $n$ points in $\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate $f(n)$ - in particular, is it true that $f(n)\leq n^{3+o(1)}$ ?

Status. Open.

Source. erdosproblems.com/1087, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1087, https://www.erdosproblems.com/1087.

References.

[ErPu71] Erdős, Paul and Purdy, George, Some extremal problems in geometry. J. Combinatorial Theory Ser. A (1971), 246-252.

Formalization. None recorded.

Progress

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Known Results

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Linked library material

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Linked from (3)

Distance Problems Erdős–Purdy: Some extremal problems in geometry Question (p. 251): how many four-point subsets of n planar points have a repeated distance

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