Let $x_{1}, \dots, x_{n} \in R^{2}$ be such that no circle whose centre is one of the $x_{i}$ contains three other points. Are there at least $(1 + c) \frac{n}{2}$ distinct distances determined between the $x_{i}$ , for some constant $c > 0$ and all $n$ sufficiently large?

Worked, still open.

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

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

Let $D$ be the number of distinct distances among the (\binom n2) pairs.

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

formal

AMS 5 52 · open (literature)

theorem erdos_655 :
    answer(sorry) ↔ ∃ c > (0 : ℝ), ∀ᶠ n in atTop, ∀ (X : Finset ℝ²), #X = n → IsValid X →
      (1 + c) * n / 2 ≤ distinctDistances X

formal-conjectures/655.lean ↗

status

open

evidence

formal

status

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