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Erdős ProblemsCombinatorics

Erdős Problem 653

Let x1,…,xn∈R2x_1,\ldots,x_n\in \mathbb{R}^2 and let R(xi)=#{∣xj−xi∣:j≠i}R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\}, where the points are ordered such that R(x1)≤⋯≤R(xn).R(x_1)\leq \cdots \leq R(x_n). Let g(n)g(n) be the maximum number of distinct values the R(xi)R(x_i) can take. Is it true that...

CombinatoricsAMS 5AMS 52

Mathematical statement

Let x1,…,xn∈R2x_1,\ldots,x_n\in \mathbb{R}^2 and let R(xi)=#{∣xj−xi∣:j≠i}R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\}, where the points are ordered such that R(x1)≤⋯≤R(xn).R(x_1)\leq \cdots \leq R(x_n). Let g(n)g(n) be the maximum number of distinct values the R(xi)R(x_i) can take. Is it true that g(n)≥(1−o(1))ng(n) \geq (1-o(1))n?

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

erdos_653

Canonical source
Complete statement target, proof intentionally absentLean 4
theorem erdos_653 : answer(sorry) ↔ ∃ o : ℕ → ℝ, o =o[atTop] (1 : ℕ → ℝ) ∧    ∀ᶠ n in atTop, (1 - o n) * n ≤ maximalDistinctDistancesFrom n := by  sorry
Statement source
Formal Conjectures
Lean version
v4.27.0
Placeholder
Present; no proof artifact
Source evidence
Pinned source index
Fidelity review
Community formulation

References

Pinned Formal Conjectures sourceformalizationerdosproblems.com/653background