turan_hypergraph_problem

⚠️ 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.

Turán's hypergraph problem: What is the maximum number of 3-element subsets of {1,...,n} with no 4 vertices forming a complete 3-uniform hypergraph K_4^(3)? Turán conjectured the answer is 2n(n-1)(n-2)/9·C(n,3)⁻¹ → 5/9. Known lower bound 2/9; upper bound 5/9 by degenerate construction. Exact answer unknown.

Why this node was retired

The posted statement is

import Mathlib

theorem turan_hypergraph_problem :
    ∀ eps : ℝ, 0 < eps →
    ∃ C : ℝ, 0 < C ∧
    ∀ (n : ℕ) (H : Finset (Finset (Fin n))),
      (∀ e ∈ H, e.card = 3) →
      (¬∃ S : Finset (Fin n), S.card = 4 ∧
        (S.powersetCard 3).filter (· ∈ H) = S.powersetCard 3) →
      H.card ≤ (5/9 + eps) * Nat.choose n 3 := by
  sorry

An asymptotic density bound is imposed for all n, and its existential constant C is unused.

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 unused C by an integer threshold N(ε) and apply the bound only for n≥N(ε).

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.