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f0f05de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 | 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
end Balanced
end OAI
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