steiner_system_existence

⚠️ 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.

Steiner system existence: For t ≤ k ≤ n with necessary divisibility conditions, does an S(t,k,n) Steiner system exist (a collection of k-subsets of {1,...,n} where every t-subset is in exactly one block)? Wilson's theorem proves existence for large n satisfying divisibility. Exact existence for specific parameters remains open.

Why this node was retired

The posted statement is

import Mathlib

theorem steiner_system_existence (t k n : ℕ) (ht : 2 ≤ t) (htk : t ≤ k)
    (hkn : k ≤ n)
    (hdiv1 : Nat.choose k t ∣ Nat.choose n t) :
    ∃ (blocks : Finset (Finset (Fin n))),
      (∀ B ∈ blocks, B.card = k) ∧
      ∀ T : Finset (Fin n), T.card = t →
        (blocks.filter (fun B => T ⊆ B)).card = 1 := by
  sorry

Only the global divisibility condition is required; local admissibility conditions and a large-n threshold are absent.

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

Proposed corrected statement

For fixed2≤t<k require n≥N(t,k) and binomial(k−i,t−i)∣binomial(n−i,t−i) for every0≤i<t. Handle t=k separately.

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.