Let $x_{1}, \dots, x_{n} \in R^{2}$ and let $R (x_{i}) = # {∣ x_{j} - x_{i} ∣ : j \neq = i}$ , where the points are ordered such that $R (x_{1}) \leq \dots \leq R (x_{n}) .$ Let $g (n)$ be the maximum number of distinct values the $R (x_{i})$ can take. Is it true that $g (n) \geq (1 - o (1)) n$ ?

Worked, still open.

geometry · open · possible · formalized (Lean) · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

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unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

What is known is that $g(n)$ is linear in $n$, but the best published lower bound is still bounded away from $n$ by a fixed constant factor:

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

AMS 5 52 · open (literature)

theorem erdos_653 : answer(sorry) ↔ ∃ o : ℕ → ℝ, o =o[atTop] (1 : ℕ → ℝ) ∧
    ∀ᶠ n in atTop, (1 - o n) * n ≤ maximalDistinctDistancesFrom n

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