erdos_ko_rado_t_intersecting

⚠️ 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–Ko–Rado theorem for t-intersecting families: Any t-intersecting family of k-element subsets of [n] has at most C(n-t, k-t) sets, provided n ≥ (t+1)(k-t+1). Proved by Frankl (1977); equality characterization for small n values involves the Hilton-Milner theorem. Extensions open.

Why this node was retired

The posted statement is

import Mathlib

theorem erdos_ko_rado_t_intersecting (n k t : ℕ) (ht : 1 ≤ t) (htk : t ≤ k) (h2k : 2 * k ≤ n + t)
    (F : Finset (Finset (Fin n)))
    (hF : ∀ A ∈ F, A.card = k)
    (hint : ∀ A ∈ F, ∀ B ∈ F, t ≤ (A ∩ B).card) :
    F.card ≤ Nat.choose (n - t) (k - t) := by
  sorry

The displayed2k≤n+t is weaker than the intended EKR size hypothesis.

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

Proposed corrected statement

Replace the size condition by n≥(t+1)(k−t+1), retaining1≤t≤k and the uniform intersection hypotheses.

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.