Lax Lean Archive

A grid minor from a strong path-of-sets system

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

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    Natural Language Statement

    Theorem

    A sufficiently long and wide strong path-of-sets system contains a prescribed square grid minor.

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    7 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.GridMinor
    2import Lax17.PathOfSets
    3
    4/-!
    5---
    6title: A grid minor from a strong path-of-sets system
    7type: theorem
    8---
    9A sufficiently long and wide strong path-of-sets system contains a
    10prescribed square grid minor.
    11-/
    12
    13namespace Lax17.StrongPathOfSetsContainsGrid
    14
    15universe u
    16
    17/-- A strong path-of-sets system of length at least `2g(g-1)` and width at
    18least `16g² + 10g` contains the `g × g` grid as a minor. -/
    19axiom strongPathOfSetsContainsGrid :
    20 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    21 (G : SimpleGraph V) {ℓ w g : ℕ},
    22 Nonempty (Lax17.PathOfSets.StrongSystem G ℓ w) →
    23 2 ≤ g →
    24 2 * g * (g - 1) ≤ ℓ →
    25 16 * g ^ 2 + 10 * g ≤ w →
    26 Lax17.GridMinor.ContainsGridMinor G g
    27
    28end Lax17.StrongPathOfSetsContainsGrid
    29
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