erdos_ulam_rational_distances

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Erdős-Ulam problem: Is there a dense subset of ℝ² with all pairwise distances rational? Conjectured no by Erdős. No such dense set is known.

Why this node was retired

The posted statement is

import Mathlib

theorem erdos_ulam_rational_distances :
    ¬∃ (pts : Set (ℝ × ℝ)),
      Dense pts ∧
      ∀ p ∈ pts, ∀ q ∈ pts, p ≠ q →
        ∃ r : ℚ, (dist p q : ℝ) = r := by
  sorry

The default distance on ℝ×ℝ is max(|Δx|,|Δy|), so rational-coordinate points give a dense set with rational distances.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Use EuclideanSpace ℝ (Fin 2), or explicitly sqrt((x₁−y₁)²+(x₂−y₂)²), in the rational-distance assertion. The genuine Erdős–Ulam question remains separate.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.