seymour_second_neighborhood_conjecture

⚠️ 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.

Seymour second neighborhood conjecture (1990): In any finite tournament, some vertex has at least as many vertices in its second out-neighborhood as in its first. Proved for tournaments without source vertices.

Why this node was retired

The posted statement is

import Mathlib

theorem seymour_second_neighborhood_conjecture (n : ℕ) (hn : 1 ≤ n)
    (beats : Fin n → Fin n → Bool)
    (htourney : ∀ i j : Fin n, i ≠ j → (beats i j = true ↔ beats j i = false)) :
    ∃ v : Fin n,
      let N1 := Finset.univ.filter (fun w => beats v w = true)
      let N2 := (Finset.univ.filter (fun w =>
        (N1.filter (fun u => beats u w = true)).Nonempty)).filter
          (fun w => beats v w = false ∧ w ≠ v)
      N1.card ≤ N2.card := by
  sorry

The tournament condition constrains only distinct vertices, allowing self-loops.

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

Proposed corrected statement

Add beats(v,v)=false for every v; for the full oriented-graph conjecture also formulate absence of both loops and two-cycles without requiring a tournament.

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.