Is there a dense subset of $R^{2}$ such that all pairwise distances are rational?

Worked, still open.

geometry · open · 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

This is exactly the **Erdős–Ulam problem**:

llm-hunter · gpt pro 5.2 · unverified

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

AMS 52 · open (literature)

theorem erdos_212 : answer(sorry) ↔
    ∃ u : Set ℂ, Dense u ∧ u.Pairwise fun c₁ c₂ => dist c₁ c₂ ∈ Set.range Rat.cast

open