Opendiscrete geometryformalizableNobody is working on it right now

Erdős Problem 1087: erdos_1087

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Added 1 Nov 2026 · not yet checked against the literature · importance 1.50

What the prompt says

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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 $f(n)\leq n^{3+o(1)}$ ?

Here a set of four points is degenerate if two of its six pairwise distances are equal, and $f(n)\leq n^{3+o(1)}$ is read as $f(n) \ll n^{3+\varepsilon}$ for every $\varepsilon > 0$ .

Lean statement

theorem erdos_1087 :
    ∀ ε > (0 : ℝ), (fun n : ℕ ↦ (f n : ℝ)) =O[atTop] fun n : ℕ ↦ (n : ℝ) ^ (3 + ε) := by
  sorry

Sources(1)

formal-conjectures: FormalConjectures/ErdosProblems/1087.lean (opens in a new tab)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/ErdosProblems/1087.lean.

Literature events(0)

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Map of approaches(0)

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Attempts(0)

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Papers on this problem(0)

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