| import Mathlib |
|
|
| namespace OAI |
|
|
| namespace Balanced |
| open scoped _root_.Balanced |
|
|
| structure PartiteHypergraph (r n : ℕ) where |
| edges : Finset (Fin r → Fin n) |
|
|
| namespace PartiteHypergraph |
|
|
| variable {r n : ℕ} |
|
|
| abbrev Vertex (r n : ℕ) := Fin r × Fin n |
|
|
| def Covers (H : PartiteHypergraph r n) (C : Finset (Vertex r n)) : Prop := |
| ∀ e ∈ H.edges, ∃ i, (i, e i) ∈ C |
|
|
| def Intersecting (H : PartiteHypergraph r n) : Prop := |
| H.edges.Nonempty ∧ ∀ e ∈ H.edges, ∀ f ∈ H.edges, ∃ i, e i = f i |
|
|
| def Nonisolated (H : PartiteHypergraph r n) : Prop := |
| ∀ i v, ∃ e ∈ H.edges, e i = v |
|
|
| def HasCoverNumber (H : PartiteHypergraph r n) (k : ℕ) : Prop := |
| (∃ C : Finset (Vertex r n), H.Covers C ∧ C.card = k) ∧ |
| ∀ C : Finset (Vertex r n), H.Covers C → k ≤ C.card |
|
|
| end PartiteHypergraph |
|
|
| def BalancedCounterexample (q : ℕ) : Prop := |
| ∃ H : PartiteHypergraph (q + 1) (q + 1), |
| H.Intersecting ∧ H.Nonisolated ∧ H.HasCoverNumber (q + 1) |
|
|
| def MainResult : Prop := |
| ∃ q₀ : ℕ, ∀ q : ℕ, q₀ ≤ q → q.Prime → BalancedCounterexample q |
|
|
| theorem main_result : MainResult := by |
| sorry |
|
|
| end Balanced |
|
|
| end OAI |
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|
