File size: 4,654 Bytes
e320e82 | 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 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | import OAI.Combinatorics.GraphThreshold.GraphSupport
import OAI.Combinatorics.GraphThreshold.GraphProbability
import OAI.Combinatorics.GraphThreshold.CopyOrbits
import OAI.Combinatorics.GraphThreshold.GraphExpectedCopies
namespace OAI
noncomputable section
open scoped Pointwise
namespace LeanBlast.SecondKahnKalai
variable {n : ℕ}
theorem mem_copyOrbit_iff_edgeSupportGraph_iso {S T : Finset (Edge n)} :
T ∈ copyOrbit (Equiv.Perm (Fin n)) S ↔
Nonempty (edgeSupportGraph S ≃g edgeSupportGraph T) := by
rw [mem_copyOrbit_iff]
constructor
· rintro ⟨σ, rfl⟩
exact ⟨edgeSupportGraph_smulIso σ S⟩
· rintro ⟨e⟩
exact exists_smul_eq_of_edgeSupportGraph_iso e
/-- Distinct edge orbits correspond bijectively to actual unlabelled subgraphs.
The target has no isolated vertices, so its image subgraph is uniquely determined
by its edges. This also covers the empty graph, with its empty vertex set. -/
def orbitSubgraphEquiv (S : Finset (Edge n)) :
OrbitCopy (Equiv.Perm (Fin n)) S ≃
{K : (⊤ : SimpleGraph (Fin n)).Subgraph //
Nonempty (edgeSupportGraph S ≃g K.coe)} where
toFun T := ⟨edgeSubgraph T.val, mem_copyOrbit_iff_edgeSupportGraph_iso.mp T.property⟩
invFun K := ⟨graphEdges K.val.spanningCoe, by
apply mem_copyOrbit_iff_edgeSupportGraph_iso.mpr
obtain ⟨e⟩ := K.property
have hK := subgraph_eq_edgeSubgraph_of_iso K.val e
change Nonempty (edgeSupportGraph S ≃g (edgeSubgraph _).coe)
rw [← hK]
exact ⟨e⟩⟩
left_inv T := by
apply Subtype.ext
exact graphEdges_graphOfEdges T.val
right_inv K := by
apply Subtype.ext
exact (subgraph_eq_edgeSubgraph_of_iso K.val K.property.some).symm
theorem orbitCount_eq_copyCount (S : Finset (Edge n)) :
orbitCount (Equiv.Perm (Fin n)) S =
(⊤ : SimpleGraph (Fin n)).copyCount (edgeSupportGraph S) := by
classical
calc
orbitCount (Equiv.Perm (Fin n)) S =
Fintype.card (OrbitCopy (Equiv.Perm (Fin n)) S) := (card_OrbitCopy _ _).symm
_ = Fintype.card {K : (⊤ : SimpleGraph (Fin n)).Subgraph //
Nonempty (edgeSupportGraph S ≃g K.coe)} := Fintype.card_congr (orbitSubgraphEquiv S)
_ = _ := by
rw [SimpleGraph.copyCount, Fintype.card_subtype]
apply congrArg Finset.card
ext K
simp only [Finset.mem_filter, Finset.mem_univ, true_and]
theorem expectedCopies_edgeSupportGraph (p : ℝ) (S : Finset (Edge n)) :
expectedCopies n p (edgeSupportGraph S) =
(orbitCount (Equiv.Perm (Fin n)) S : ℝ) * p ^ S.card := by
rw [expectedCopies_formula, edgeCount_edgeSupportGraph, orbitCount_eq_copyCount]
/-- The internal orbit constraint follows from an actual subgraph constraint
of the public expectation threshold, with no multiplicity factor. -/
theorem orbit_expectation_constraint {V : Type*} (f : V ↪ Fin n)
(H : SimpleGraph V) (q : ℝ)
(hq : ∀ F : H.Subgraph, (1 / 2 : ℝ) ≤ expectedCopies n q F.coe)
(S : Finset (Edge n)) (hS : S ⊆ graphEdges (H.map f)) :
(1 / 2 : ℝ) ≤ (orbitCount (Equiv.Perm (Fin n)) S : ℝ) * q ^ S.card := by
obtain ⟨F, ⟨e⟩⟩ := exists_subgraph_iso_edgeSupportGraph f H S hS
rw [← expectedCopies_edgeSupportGraph, expectedCopies_iso n q e]
exact hq F
theorem copyCount_pos_iff_orbit_subset {V : Type*} (f : V ↪ Fin n)
(H : SimpleGraph V) (W : Finset (Edge n)) :
0 < (graphOfEdges W).copyCount H ↔
∃ E ∈ copyOrbit (Equiv.Perm (Fin n)) (graphEdges (H.map f)), E ⊆ W := by
rw [copyCount_pos_iff_exists_smul_subset f H W]
constructor
· rintro ⟨σ, hσ⟩
exact ⟨_, (mem_copyOrbit_iff _).mpr ⟨σ, rfl⟩, hσ⟩
· rintro ⟨E, hE, hEW⟩
obtain ⟨σ, rfl⟩ := (mem_copyOrbit_iff _).mp hE
exact ⟨σ, hEW⟩
theorem exists_graphRepresentationWithConstraints {V : Type*} [Fintype V]
(n : ℕ) (H : SimpleGraph V) (hcard : Fintype.card V ≤ n) (q : ℝ)
(hq : ∀ F : H.Subgraph, (1 / 2 : ℝ) ≤ expectedCopies n q F.coe) :
∃ E : Finset (Edge n), E.card = edgeCount H ∧
(∀ W : Finset (Edge n),
(0 < (graphOfEdges W).copyCount H ↔
∃ T ∈ copyOrbit (Equiv.Perm (Fin n)) E, T ⊆ W)) ∧
(∀ S : Finset (Edge n), S ⊆ E →
(1 / 2 : ℝ) ≤ (orbitCount (Equiv.Perm (Fin n)) S : ℝ) * q ^ S.card) := 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_orbit_subset f H, orbit_expectation_constraint f H q hq⟩
end LeanBlast.SecondKahnKalai
end
end OAI
|