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Lax17.PolynomialGridMinor

Exponent 8 (×\times Polylogarithmic) Bound for the Grid-Minor Theorem

concepts/Lax17/PolynomialGridMinor.lean · lax-17

proven

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

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    Theorem

    There are positive integers KK and bb such that every finite simple graph of treewidth at least

    Kg8(log⁡2g)bK g^8 (\log_2 g)^b

    contains the g×gg \times g square grid as a minor.

    Treewidth is defined through finite tree decompositions, and minor containment uses branch sets.

    Lean source view on GitHub

    1import Mathlib.Analysis.SpecialFunctions.Log.Base
    2import Lax17.GridMinor
    3import Lax17.Treewidth
    4
    5/-!
    6---
    7title: Exponent 8 (×\times Polylogarithmic) Bound for the Grid-Minor Theorem
    8type: theorem
    9---
    10There are positive integers KK and bb such that every finite simple graph
    11of treewidth at least
    12
    13Kg8(log⁡2g)bK g^8 (\log_2 g)^b
    14
    15contains the g×gg \times g square grid as a minor.
    16
    17Treewidth is defined through finite tree decompositions, and minor
    18containment uses branch sets.
    19-/
    20
    21namespace Lax17.PolynomialGridMinor
    22
    23universe u
    24
    25/-- Exponent-eight grid-minor bound with a natural-number polylogarithmic
    26factor. -/
    27axiom polynomial_grid_minor_eight_polylog :
    28 ∃ K b : ℕ, 0 < K ∧ 0 < b ∧
    29 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    30 (G : SimpleGraph V) {g : ℕ},
    31 2 ≤ g →
    32 K * g ^ 8 * (Nat.log 2 g) ^ b ≤
    33 Lax17.Treewidth.treewidth G →
    34 Lax17.GridMinor.ContainsGridMinor G g
    35
    36end Lax17.PolynomialGridMinor
    37
    Show Proof

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