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5d5048e | 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 | import Mathlib
namespace OAI
namespace LinearListHadwiger
def ListColorable {V : Type} (G : SimpleGraph V) {Color : Type}
(L : V → Finset Color) : Prop :=
∃ c : V → Color, (∀ v, c v ∈ L v) ∧ ∀ ⦃v w⦄, G.Adj v w → c v ≠ c w
def Choosable {V : Type} (G : SimpleGraph V) (k : ℕ) : Prop :=
∀ (Color : Type) (L : V → Finset Color), (∀ v, k ≤ (L v).card) → ListColorable G L
noncomputable def listChromaticNumber {V : Type} (G : SimpleGraph V) : ℕ :=
sInf {k : ℕ | Choosable G k}
def HasCliqueMinor {V : Type} (G : SimpleGraph V) (t : ℕ) : Prop :=
∃ B : Fin t → Set V,
(∀ i, (G.induce (B i)).Connected) ∧
(∀ i j, i ≠ j → Disjoint (B i) (B j)) ∧
(∀ i j, i ≠ j → ∃ v ∈ B i, ∃ w ∈ B j, G.Adj v w)
noncomputable def hadwigerNumber {V : Type} (G : SimpleGraph V) : ℕ :=
sSup {t : ℕ | HasCliqueMinor G t}
def MainStatement : Prop :=
∃ C : ℕ, 1 ≤ C ∧ ∀ (V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V),
listChromaticNumber G ≤ C * hadwigerNumber G
end LinearListHadwiger
end OAI
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