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Lax17.StrongPathOfSetsFromTreewidth

A strong path-of-sets system from treewidth

concepts/Lax17/StrongPathOfSetsFromTreewidth.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 strong path-of-sets system of prescribed length and width.

    Lean source view on GitHub

    1import Mathlib.Data.Nat.Log
    2import Lax17.PathOfSets
    3import Lax17.Treewidth
    4
    5/-!
    6---
    7title: A strong path-of-sets system from treewidth
    8type: theorem
    9---
    10Sufficiently large treewidth produces a strong path-of-sets system of
    11prescribed length and width.
    12-/
    13
    14namespace Lax17.StrongPathOfSetsFromTreewidth
    15
    16universe u
    17
    18/-- Sufficiently large treewidth produces a strong path-of-sets system of
    19prescribed length and width. -/
    20axiom strongPathOfSetsFromTreewidth :
    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 Nonempty (Lax17.PathOfSets.StrongSystem G ℓ w)
    30
    31end Lax17.StrongPathOfSetsFromTreewidth
    32
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