| import OAI.Combinatorics.GraphThreshold.ThresholdBounds |
| import OAI.Combinatorics.GraphThreshold.GraphCopies |
| import OAI.Combinatorics.GraphThreshold.GraphHierarchy |
| import OAI.Combinatorics.GraphThreshold.BernoulliMonotonicity |
| import OAI.Combinatorics.GraphThreshold.TreeCovering |
|
|
| namespace OAI |
|
|
| /-! |
| # The second Kahn–Kalai graph-threshold bounds |
|
|
| The graph-to-orbit bridge counts actual unlabelled subgraphs. The hierarchy is |
| constructed from the constraints at the attained expectation threshold, and |
| covering one of its paths yields a copy of the original graph. |
| -/ |
|
|
| noncomputable section |
|
|
| namespace LeanBlast.SecondKahnKalai |
|
|
| universe u |
|
|
| def FullStatement : Prop := SecondKahnKalaiStatement.{u} |
|
|
| theorem secondKahnKalaiBounds (n : ℕ) (V : Type u) [Fintype V] |
| (H : SimpleGraph V) (hn : 2 ≤ n) (hcard : Fintype.card V ≤ n) |
| (hedge : 1 ≤ edgeCount H) : SecondKahnKalaiBounds n H := by |
| classical |
| let q := expectationThreshold n H |
| have hq : 0 < q := expectationThreshold_pos n H hcard hedge |
| obtain ⟨E, hEcard, hEevent, hEconstraint⟩ := |
| exists_graphRepresentationWithConstraints n H hcard q |
| (expectationThreshold_attained n H hcard).2 |
| have hEnonempty : E.Nonempty := Finset.card_pos.mp (by omega) |
| obtain ⟨k, m, T, hk, hdisjoint, hcapacity, hspread, hpositive, hgrowth, hlast, hpath⟩ := |
| exists_spread_orbit_tree (Equiv.Perm (Fin n)) E q hq hEnonempty |
| (fun S hS _ => hEconstraint S hS) |
| have hm : 0 < m (k - 1) := hpositive (k - 1) (by omega) |
| have hmh : m (k - 1) ≤ edgeCount H := by omega |
| let p := min 1 (2048 * Real.exp 50 * q * (1 + logTwo (edgeCount H : ℝ))) |
| let pTree := min 1 (16 * Real.exp 50 * (128 * q) * |
| (1 + Real.logb 2 (m (k - 1) : ℝ))) |
| have hp : p ∈ Set.Icc 0 1 := graphThresholdParameter_mem_unitInterval q _ hq.le hedge |
| have hpTree : pTree ∈ Set.Icc 0 1 := treeSamplingParameter_mem_unitInterval q _ hq.le hm |
| have hpTree_le : pTree ≤ p := |
| treeSamplingParameter_le_graphParameter q _ _ hq.le hm hmh |
| have hcovered : (2 / 3 : ℝ) ≤ bernoulliProbability pTree T.CoveredBy := |
| tree_covering T hk hdisjoint m hcapacity hpositive hgrowth (128 * q) |
| (by positivity) hspread |
| have hincrease : bernoulliProbability pTree T.CoveredBy ≤ |
| bernoulliProbability p T.CoveredBy := by |
| apply bernoulliProbability_mono_parameter hpTree.1 hpTree_le hp.2 T.CoveredBy |
| rintro W W' hWW' ⟨path, hpathW⟩ |
| exact ⟨path, hpathW.trans hWW'⟩ |
| have hevent : bernoulliProbability p T.CoveredBy ≤ containmentProbability n p H := by |
| change (bernoulliLaw p hp.1 hp.2).prob T.CoveredBy ≤ |
| (bernoulliLaw p hp.1 hp.2).prob (fun W => 0 < (graphOfEdges W).copyCount H) |
| apply FiniteLaw.prob_mono |
| rintro W ⟨path, hpathW⟩ |
| exact (hEevent W).mpr ⟨T.pathUnion path, hpath path, hpathW⟩ |
| have hpc : criticalThreshold n H ≤ p := by |
| apply criticalThreshold_le_of_probability n H hp |
| exact (by norm_num : (1 / 2 : ℝ) ≤ 2 / 3).trans |
| (hcovered.trans (hincrease.trans hevent)) |
| exact ⟨hpc, hpc.trans (graphThresholdParameter_le_vertexBound n H q hn hcard hedge hq.le)⟩ |
|
|
| theorem secondKahnKalai : FullStatement.{u} := by |
| intro n V inst H hn hcard hedge |
| exact secondKahnKalaiBounds n V H hn hcard hedge |
|
|
| end LeanBlast.SecondKahnKalai |
|
|
| end |
|
|
| end OAI |
|
|
