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Lax17.ExpanderGrid

Grid minors from separator expansion

concepts/Lax17/ExpanderGrid.lean · lax-17

proven

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    Concept map

    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

    Evidence

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    Theorem

    A graph with no small balanced separator contains every square grid whose vertices-plus-edges complexity fits the explicit Theorem 8.1 budget.

    Lean source view on GitHub

    1import Mathlib.Data.Nat.Log
    2import Lax17.Degree
    3import Lax17.Expansion
    4import Lax17.GridMinor
    5
    6/-!
    7---
    8title: Grid minors from separator expansion
    9type: theorem
    10---
    11A graph with no small balanced separator contains every square grid whose
    12vertices-plus-edges complexity fits the explicit Theorem 8.1 budget.
    13-/
    14
    15namespace Lax17.ExpanderGrid
    16
    17universe u
    18
    19/-- The expander-to-grid handoff in the no-small-balanced-separator form. -/
    20axiom expanderContainsGrid :
    21 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    22 (G : SimpleGraph V) (d g : ℕ),
    23 2 ≤ Fintype.card V →
    24 Lax17.Expansion.NoSmallBalancedSeparator G d →
    25 ((3 * (d + 1) * (15 * (d + 1))) * 8) *
    26 (5 * g ^ 2) * Nat.log 2 (Fintype.card V) ≤
    27 Fintype.card V →
    28 Lax17.GridMinor.ContainsGridMinor G g
    29
    30end Lax17.ExpanderGrid
    31
    Show Proof

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