borsuk_partition_conjecture

⚠️ Retired — specification defect

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

Borsuk's problem (1932): Can every bounded set in ℝᵈ be partitioned into d+1 pieces each with strictly smaller diameter? True for d ≤ 3. Counterexamples exist for d ≥ 64 (Kahn–Kalai 1993). The problem remains open for 4 ≤ d ≤ 63.

Why this node was retired

The posted statement is

import Mathlib

theorem borsuk_partition_conjecture (d : ℕ) (hd : 1 ≤ d) :
    ∀ (S : Finset (EuclideanSpace ℝ (Fin d))),
      (∃ R : ℝ, 0 < R ∧ ∀ x ∈ S, ∀ y ∈ S, dist x y ≤ R) →
      ∃ R : ℝ, (∀ x ∈ S, ∀ y ∈ S, dist x y = R → False) ∨
      ∃ (parts : Fin (d + 1) → Finset (EuclideanSpace ℝ (Fin d))),
        (∀ x ∈ S, ∃ i, x ∈ parts i) ∧
        ∀ i, ∃ diam : ℝ, ∀ x ∈ parts i, ∀ y ∈ parts i,
          dist x y ≤ diam ∧
          ∀ x' ∈ S, ∀ y' ∈ S, dist x' y' ≤ diam := by
  sorry

An unrelated negative R makes the first disjunct true; R is not required to be the diameter and the partition is never used.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

For a bounded subset S of Euclidean ℝⁿ with diameter D>0, ask for a cover by n+1 subsets Sᵢ with diam(Sᵢ)<D. Specify a dimension/regime where the statement is true; the unrestricted Borsuk conjecture is false. Remove the unrelated disjunct.

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.