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Lax17.HairyPathOfSetsFromTreewidth

A hairy path-of-sets system from treewidth

concepts/Lax17/HairyPathOfSetsFromTreewidth.lean · lax-17

proven

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

    Evidence

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    Theorem

    Sufficiently large treewidth produces a hairy strong path-of-sets system.

    Lean source view on GitHub

    1import Mathlib.Data.Nat.Log
    2import Lax17.Degree
    3import Lax17.PathOfSets
    4import Lax17.Treewidth
    5
    6/-!
    7---
    8title: A hairy path-of-sets system from treewidth
    9type: theorem
    10---
    11Sufficiently large treewidth produces a hairy strong path-of-sets system.
    12-/
    13
    14namespace Lax17.HairyPathOfSetsFromTreewidth
    15
    16universe u
    17
    18/-- Sufficiently large treewidth produces a hairy strong path-of-sets
    19system. -/
    20axiom hairyPathOfSetsFromTreewidth :
    21 ∃ c d : ℕ, 0 < c ∧ 0 < d ∧
    22 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    23 (G : SimpleGraph V) {ℓ w k : ℕ},
    24 1 < ℓ →
    25 1 < w →
    26 1 < k →
    27 k ≤ Lax17.Treewidth.treewidth G →
    28 c * w * ℓ ^ 50 * (Nat.log 2 k) ^ d < k →
    29 ∃ H : SimpleGraph V,
    30 H ≤ G ∧
    31 Lax17.Degree.MaximumAtMost H 3 ∧
    32 Nonempty (Lax17.PathOfSets.HairySystem H ℓ w)
    33
    34end Lax17.HairyPathOfSetsFromTreewidth
    35
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