File size: 3,294 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 | 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
|