| import OAI.Combinatorics.CliqueFree.MaxDegree |
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| namespace OAI |
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| noncomputable section |
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| open scoped BigOperators |
| open Finset |
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| attribute [local instance] Classical.propDecidable |
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| namespace CliqueFreeLog |
| open CliqueFreeIndependence.WeightedGraph |
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| lemma sum_degrees_real {V : Type*} [Fintype V] (G : SimpleGraph V) : |
| (∑ v, (G.degree v : ℝ)) = 2 * (G.edgeFinset.card : ℝ) := by |
| classical |
| exact_mod_cast G.sum_degrees_eq_twice_card_edges |
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| / |
| average degree, with a constant uniform in graph size and average degree. -/ |
| theorem logarithmic_independence_bound (r : ℕ) (hr : 4 ≤ r) : |
| ∃ c : ℝ, 0 < c ∧ |
| ∀ {V : Type} [Fintype V] (G : SimpleGraph V), |
| G.CliqueFree r → 2 ≤ averageDegree G → |
| c * (Fintype.card V : ℝ) * Real.log (averageDegree G) / averageDegree G ≤ |
| (G.indepNum : ℝ) := by |
| classical |
| obtain ⟨B,hB,htri⟩ := exists_optimizer_triangle_bound.{0} (r-2) |
| let D : ℝ := 3+3*B/2 |
| have hD : 0 < D := by dsimp [D]; positivity |
| refine ⟨1/(16*D),by positivity,?_⟩ |
| intro V _ G hf hd |
| let d : ℝ := averageDegree G |
| have hd2 : 2 ≤ d := hd |
| have hdpos : 0 < d := by linarith |
| have hn : (Fintype.card V : ℝ) ≠ 0 := by |
| intro hn |
| have hz : averageDegree G = 0 := by simp [averageDegree,hn] |
| rw [hz] at hd |
| norm_num at hd |
| have hsum : (∑ v, (G.degree v : ℝ)) = (Fintype.card V : ℝ)*d := by |
| rw [sum_degrees_real] |
| dsimp [d, averageDegree] |
| field_simp |
| let A : Finset V := univ.filter fun v ↦ (G.degree v : ℝ) ≤ 2*d |
| have hAcard : (Fintype.card V : ℝ)/2 ≤ (Fintype.card (A : Set V) : ℝ) := by |
| have he : Fintype.card (A : Set V) = A.card := Fintype.card_coe A |
| rw [he] |
| exact degree_cutoff_card G hdpos hsum |
| have hk : r-2+2 = r := Nat.sub_add_cancel (by omega) |
| have hfree : (G.induce (A : Set V)).CliqueFree (r-2+2) := by |
| rw [hk] |
| exact (G.cliqueFree_induce_iff _ _).2 hf.cliqueFreeOn |
| have hdeg (v : (A : Set V)) : ((G.induce (A : Set V)).degree v : ℝ) ≤ 2*d := by |
| have hv : (G.degree v : ℝ) ≤ 2*d := (mem_filter.1 v.property).2 |
| exact (Nat.cast_le.2 (degree_induce_le G A v)).trans hv |
| have hmax := maximum_degree_independence (r-2) hB.le (by linarith : 3 ≤ 2*d) |
| htri (G.induce (A : Set V)) hfree hdeg |
| have hi : ((G.induce (A : Set V)).indepNum : ℝ) ≤ G.indepNum := |
| Nat.cast_le.2 (indepNum_induce_le G A) |
| have hlog : Real.log d ≤ Real.log (2*d) := |
| Real.log_le_log hdpos (by linarith) |
| have hlog0 : 0 ≤ Real.log d := Real.log_nonneg (by linarith) |
| have hnum : (Fintype.card V : ℝ)*Real.log d/2 ≤ |
| (Fintype.card (A : Set V) : ℝ)*Real.log (2*d) := by |
| have h1 := mul_le_mul_of_nonneg_right hAcard hlog0 |
| have h2 := mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg (α := ℝ) (Fintype.card (A : Set V))) |
| nlinarith |
| have hden : 0 < 8*D*d := by positivity |
| have hcompare := div_le_div_of_nonneg_right hnum hden.le |
| change (Fintype.card (A : Set V) : ℝ)*Real.log (2*d)/(4*D*(2*d)) ≤ _ at hmax |
| have heq : (1/(16*D))*(Fintype.card V : ℝ)*Real.log d/d = |
| ((Fintype.card V : ℝ)*Real.log d/2)/(8*D*d) := by |
| field_simp |
| ring |
| change (1/(16*D))*(Fintype.card V : ℝ)*Real.log d/d ≤ _ |
| rw [heq] |
| apply hcompare.trans |
| have hdeneq : 8*D*d = 4*D*(2*d) := by ring |
| rw [hdeneq] |
| exact hmax.trans hi |
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| end CliqueFreeLog |
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| end |
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| end OAI |
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