File size: 2,527 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 | import Mathlib.Data.Fintype.Powerset
import Mathlib.Basic.Real.Basic
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Combinatorics.SimpleGraph.Copy
import Mathlib.Analysis.SpecialFunctions.Log.Basic
namespace OAI
noncomputable section
universe u
namespace LeanBlast.SecondKahnKalai
def bernoulliWeight {X : Type u} [Fintype X] [DecidableEq X]
(p : ℝ) (W : Finset X) : ℝ :=
∏ x, if x ∈ W then p else 1 - p
def bernoulliExpectation {X : Type u} [Fintype X] [DecidableEq X]
(p : ℝ) (f : Finset X → ℝ) : ℝ :=
∑ W, bernoulliWeight p W * f W
def bernoulliProbability {X : Type u} [Fintype X] [DecidableEq X]
(p : ℝ) (P : Finset X → Prop) : ℝ := by
classical
exact ∑ W, if P W then bernoulliWeight p W else 0
/-- The unordered non-loop edges of the complete graph on `n` vertices. -/
abbrev Edge (n : ℕ) := {e : Sym2 (Fin n) // ¬e.IsDiag}
/-- A set of edges, interpreted as an ordinary graph on all `n` vertices. -/
def graphOfEdges {n : ℕ} (W : Finset (Edge n)) : SimpleGraph (Fin n) :=
SimpleGraph.fromEdgeSet (Subtype.val '' (W : Set (Edge n)))
/-- The number of unordered edges, each counted once. -/
def edgeCount {V : Type u} (H : SimpleGraph V) : ℕ := Nat.card H.edgeSet
/-- Expected number of actual unlabelled non-induced copies in `G(n,p)`. -/
def expectedCopies {V : Type u} (n : ℕ) (p : ℝ) (H : SimpleGraph V) : ℝ :=
bernoulliExpectation p (fun W : Finset (Edge n) =>
((graphOfEdges W).copyCount H : ℝ))
/-- Probability that `G(n,p)` contains at least one non-induced copy. -/
def containmentProbability {V : Type u} (n : ℕ) (p : ℝ)
(H : SimpleGraph V) : ℝ :=
bernoulliProbability p (fun W : Finset (Edge n) =>
0 < (graphOfEdges W).copyCount H)
def criticalThreshold {V : Type u} (n : ℕ) (H : SimpleGraph V) : ℝ :=
sInf {p : ℝ | p ∈ Set.Icc 0 1 ∧ (1 / 2 : ℝ) ≤ containmentProbability n p H}
def expectationThreshold {V : Type u} (n : ℕ) (H : SimpleGraph V) : ℝ :=
sInf {p : ℝ | p ∈ Set.Icc 0 1 ∧
∀ F : H.Subgraph, (1 / 2 : ℝ) ≤ expectedCopies n p F.coe}
def logTwo (x : ℝ) : ℝ := Real.log x / Real.log 2
def SecondKahnKalaiBounds {V : Type u} (n : ℕ) (H : SimpleGraph V) : Prop :=
criticalThreshold n H ≤
min 1 (2048 * Real.exp 50 * expectationThreshold n H *
(1 + logTwo (edgeCount H : ℝ))) ∧
criticalThreshold n H ≤
6144 * Real.exp 50 * expectationThreshold n H * logTwo (n : ℝ)
end LeanBlast.SecondKahnKalai
end
end OAI
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