@@ -3,8 +3,8 @@ Copyright (c) 2023 Yaël Dillies. All rights reserved.
33Released under Apache 2.0 license as described in the file LICENSE.
44Authors: Yaël Dillies
55-/
6- import LeanCamCombi.ExtrProbCombi.BinomialRandomGraph
76import LeanCamCombi.Mathlib.Combinatorics.SimpleGraph.Density
7+ import LeanCamCombi.Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs
88import Mathlib.Analysis.SpecialFunctions.Pow.Real
99import Mathlib.Combinatorics.SimpleGraph.Copy
1010import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
@@ -16,25 +16,23 @@ This file proves Bollobás' lemma on graph containment.
1616-/
1717
1818open Asymptotics Filter MeasureTheory
19- open scoped MeasureTheory ENNReal NNReal SimpleGraph Topology
20-
21- variable {α β Ω : Type *} [Fintype β] {G : ℕ → Ω → SimpleGraph α} (H : SimpleGraph β)
22- [Fintype H.edgeSet] [∀ n ω, DecidableRel (G n ω).Adj] [MeasurableSpace Ω] (μ : Measure Ω)
23- [IsProbabilityMeasure μ] {p : ℕ → ℝ≥0 } (hG : ∀ n, IsBinomialRandomGraph (G n) (p n) μ)
19+ open scoped MeasureTheory ENNReal NNReal SimpleGraph Topology unitInterval
2420
2521namespace SimpleGraph
22+ variable {V W Ω : Type *} [Fintype W] {G : ℕ → Ω → SimpleGraph V} (H : SimpleGraph W)
23+ [Fintype H.edgeSet] [∀ n ω, DecidableRel (G n ω).Adj] [MeasurableSpace Ω] (μ : Measure Ω)
24+ [IsProbabilityMeasure μ] {p : ℕ → I} (hG : ∀ n, IsBinomialRandom (G n) (p n) μ)
2625
27- /-- **Bollobás' Graph Containment Lemma, zero statement** -/
26+ /-- **Bollobás' Graph Containment Lemma** , zero statement -/
2827lemma zero_statement
2928 (hp : (fun n ↦ p n : ℕ → ℝ) =o[atTop] (fun n ↦ n ^ (-H.maxEdgeSubdensity⁻¹ : ℝ) : ℕ → ℝ)) :
3029 Tendsto (fun n ↦ μ {ω | H ⊑ G n ω}) atTop (𝓝 0 ) :=
3130 sorry
3231
33- /-- **Bollobás' Graph Containment Lemma, one statement** -/
32+ /-- **Bollobás' Graph Containment Lemma** , one statement -/
3433lemma one_statement
3534 (hp : (fun n ↦ n ^ (-H.maxEdgeSubdensity⁻¹ : ℝ) : ℕ → ℝ) =o[atTop] (fun n ↦ p n : ℕ → ℝ)) :
3635 Tendsto (fun n ↦ μ {ω | H ⊑ G n ω}) atTop (𝓝 1 ) :=
3736 sorry
3837
3938end SimpleGraph
40-
0 commit comments