alspach_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.

Alspach's conjecture (1981): For any collection of cycle lengths m₁,...,mₖ with each mᵢ ≥ 3, mᵢ ≤ n, and Σmᵢ = n(n-1)/2, the complete graph Kₙ can be decomposed into cycles of these lengths. Proved for odd n ≤ 20000 and even n; the full conjecture for all n was proved by Bryant–Horsley–Pettersson (2014). This is now a theorem.

Why this node was retired

The posted statement is

import Mathlib

theorem alspach_conjecture (n : ℕ) (hn : 3 ≤ n)
    (k : ℕ) (cycles : Fin k → ℕ)
    (hcycles_len : ∀ i, 3 ≤ cycles i ∧ cycles i ≤ n)
    (hsum : ∑ i, cycles i = n * (n - 1) / 2) :
    ∃ (decomp : Fin k → Finset (Sym2 (Fin n))),
      let KN : SimpleGraph (Fin n) := ⊤
      (∀ i, decomp i ⊆ KN.edgeFinset) ∧
      (∀ e ∈ KN.edgeFinset, ∃! i, e ∈ decomp i) ∧
      ∀ i, (decomp i).card = cycles i := by
  sorry

The parts of the edge partition need not be cycles.

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

Require each part to be the edge set of a simple cycle of its assigned length. For the standard complete-graph formulation also use odd n, or handle a removed perfect matching in the even-n case.

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.