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A well-linked set from treewidth

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

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    Natural Language Statement

    Theorem

    Large treewidth yields a proportionally large node-well-linked set.

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

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax17.Linkedness
    2import Lax17.Treewidth
    3
    4/-!
    5---
    6title: A well-linked set from treewidth
    7type: theorem
    8---
    9Large treewidth yields a proportionally large node-well-linked set.
    10-/
    11
    12namespace Lax17.NodeWellLinkedSetFromTreewidth
    13
    14universe u
    15
    16/-- Large treewidth yields a large node-well-linked vertex set. -/
    17axiom nodeWellLinkedSetFromTreewidth :
    18 ∃ c : ℕ, 0 < c ∧
    19 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    20 (G : SimpleGraph V) {k : ℕ},
    21 c * k ≤ Lax17.Treewidth.treewidth G →
    22 ∃ X : Finset V,
    23 k ≤ X.card ∧
    24 Lax17.Linkedness.NodeWellLinkedIn G Finset.univ X
    25
    26end Lax17.NodeWellLinkedSetFromTreewidth
    27
    Show Proof
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