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Restore geometric Kneser graph isomorphisms #178

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mo271 merged 1 commit into from
Oct 27, 2026
Merged

Restore geometric Kneser graph isomorphisms #178

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15 changes: 13 additions & 2 deletions FormalBook/Chapter_43.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -461,12 +461,23 @@ def GeometricKneserGraph {d : ℕ} (P : Finset (EuclideanSpace ℝ (Fin d))) (k
def isoGeometricKneserGraphVertices {d : ℕ} (P : Finset (EuclideanSpace ℝ (Fin d))) (k : ℕ) :
{s : Finset (EuclideanSpace ℝ (Fin d)) // s ⊆ P ∧ s.card = k} ≃
{s : Finset (Fin P.card) // s.card = k} := by
sorry
let e := (Equiv.finsetSubtypeComm (· ∈ P)).symm.trans P.equivFin.finsetCongr
exact (Equiv.subtypeSubtypeEquivSubtypeInter (· ⊆ P) (fun s => s.card = k)).symm.trans
(e.subtypeEquiv (by intro s; simp [e, Equiv.finsetSubtypeComm]))

/-- The geometric Kneser graph on `P` is isomorphic to the standard Kneser graph `K(|P|, k)`. -/
def isoGeometricKneserGraph {d : ℕ} (P : Finset (EuclideanSpace ℝ (Fin d))) (k : ℕ) :
GeometricKneserGraph P k ≃g KneserGraph P.card k := by
sorry
refine (show KneserGraph P.card k ≃g GeometricKneserGraph P k from
{ toEquiv := (isoGeometricKneserGraphVertices P k).symm
map_rel_iff' := ?_ }).symm
intro v w
simp only [GeometricKneserGraph, KneserGraph, SimpleGraph.fromRel_adj,
Equiv.apply_eq_iff_eq, ne_eq, or_self, disjoint_comm]
change (v ≠ w ∧
Disjoint ((v.val.map P.equivFin.symm.toEmbedding).map (Function.Embedding.subtype _))
((w.val.map P.equivFin.symm.toEmbedding).map (Function.Embedding.subtype _))) ↔ _
simp only [Finset.disjoint_map]

/- The chromatic number of the geometric Kneser graph on `P` is strictly greater than `d+1`. -/
theorem chromatic_number_geometric_kneser_gt {k d : ℕ} (hk : 1 ≤ k)
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