| import OAI.Combinatorics.GraphThreshold.GraphEncoding |
| import OAI.Combinatorics.GraphThreshold.EmbeddingExtension |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped Pointwise |
|
|
| namespace LeanBlast.SecondKahnKalai |
|
|
| variable {n : ℕ} {V : Type*} |
|
|
| theorem card_mappedGraphEdges [Fintype V] (f : V ↪ Fin n) (H : SimpleGraph V) : |
| (graphEdges (H.map f)).card = edgeCount H := by |
| classical |
| rw [card_graphEdges, edgeCount_eq_card_edgeFinset, edgeCount_eq_card_edgeFinset] |
| exact SimpleGraph.card_edgeFinset_map f H |
|
|
| theorem mapped_copy_le {U W : Type*} {H : SimpleGraph U} {G : SimpleGraph W} |
| (c : H.Copy G) : H.map c.toEmbedding ≤ G := by |
| intro a b hab |
| obtain ⟨v, w, hvw, rfl, rfl⟩ := (SimpleGraph.map_adj c.toEmbedding H a b).mp hab |
| exact c.toHom.map_adj hvw |
|
|
| /-- Containment of a graph is equivalent to containing a permuted copy of its |
| edges. The embedding carries *all* vertices, so isolated vertices are retained. -/ |
| theorem isContained_iff_exists_smul_subset (f : V ↪ Fin n) (H : SimpleGraph V) |
| (W : Finset (Edge n)) : |
| H.IsContained (graphOfEdges W) ↔ |
| ∃ σ : Equiv.Perm (Fin n), σ • graphEdges (H.map f) ⊆ W := by |
| constructor |
| · rintro ⟨c⟩ |
| obtain ⟨σ, hσ⟩ := exists_perm_comp_embedding f c.toEmbedding |
| refine ⟨σ, graphOfEdges_le_iff.mp ?_⟩ |
| rw [graphOfEdges_smul, graphOfEdges_graphEdges, SimpleGraph.map_map] |
| have heq : (fun v => σ.toEmbedding (f v)) = c.toEmbedding := by |
| funext v |
| exact hσ v |
| change H.map (fun v => σ.toEmbedding (f v)) ≤ graphOfEdges W |
| rw [heq] |
| exact mapped_copy_le c |
| · rintro ⟨σ, hσ⟩ |
| have hf : H.IsContained (H.map f) := |
| ⟨(SimpleGraph.Embedding.map f H).toCopy⟩ |
| have hperm : (H.map f).IsContained (graphOfEdges (σ • graphEdges (H.map f))) := by |
| rw [graphOfEdges_smul, graphOfEdges_graphEdges] |
| exact ⟨(SimpleGraph.Iso.map σ _).toCopy⟩ |
| exact hf.trans (hperm.trans (SimpleGraph.IsContained.of_le (graphOfEdges_mono hσ))) |
|
|
| theorem copyCount_pos_iff_exists_smul_subset (f : V ↪ Fin n) (H : SimpleGraph V) |
| (W : Finset (Edge n)) : |
| 0 < (graphOfEdges W).copyCount H ↔ |
| ∃ σ : Equiv.Perm (Fin n), σ • graphEdges (H.map f) ⊆ W := |
| SimpleGraph.copyCount_pos.trans (isContained_iff_exists_smul_subset f H W) |
|
|
| theorem exists_graphRepresentation [Fintype V] (n : ℕ) (H : SimpleGraph V) |
| (hcard : Fintype.card V ≤ n) : |
| ∃ E : Finset (Edge n), E.card = edgeCount H ∧ |
| ∀ W : Finset (Edge n), |
| (0 < (graphOfEdges W).copyCount H ↔ |
| ∃ σ : Equiv.Perm (Fin n), σ • E ⊆ W) := by |
| obtain ⟨f⟩ := Function.Embedding.nonempty_of_card_le |
| (show Fintype.card V ≤ Fintype.card (Fin n) by simpa using hcard) |
| exact ⟨graphEdges (H.map f), card_mappedGraphEdges f H, |
| copyCount_pos_iff_exists_smul_subset f H⟩ |
|
|
| end LeanBlast.SecondKahnKalai |
|
|
| end |
|
|
| end OAI |
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|
