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Strong tree-of-sets construction

Lax17.StrongTreeOfSetsConstruction · concepts/Lax17/StrongTreeOfSetsConstruction.lean · lax-17

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    Natural Language Statement

    Theorem

    A sufficiently large node-well-linked set supports a strong subcubic tree-of-sets system.

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    6 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    1import Mathlib.Data.Nat.Log
    2import Lax17.TreeOfSets
    3
    4/-!
    5---
    6title: Strong tree-of-sets construction
    7type: theorem
    8---
    9A sufficiently large node-well-linked set supports a strong subcubic
    10tree-of-sets system.
    11-/
    12
    13namespace Lax17.StrongTreeOfSetsConstruction
    14
    15universe u
    16
    17/-- A sufficiently large node-well-linked set supports a strong subcubic
    18tree-of-sets system. -/
    19axiom strongTreeOfSetsConstruction :
    20 ∃ c d p : ℕ, 0 < c ∧ 0 < d ∧ 0 < p ∧
    21 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    22 (G : SimpleGraph V) (X : Finset V) {m w x Δ : ℕ},
    23 1 < m →
    24 1 < w →
    25 1 < x →
    26 Lax17.Degree.MaximumAtMost G Δ →
    27 X.card = x →
    28 Lax17.Linkedness.NodeWellLinkedIn G Finset.univ X →
    29 c * w * m ^ 24 * Δ ^ p *
    30 (Nat.log 2 x) ^ d < x →
    31 Nonempty (Lax17.TreeOfSets.StrongSystem G m w)
    32
    33end Lax17.StrongTreeOfSetsConstruction
    34
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