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A low-degree well-linked core

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

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    Natural Language Statement

    Theorem

    A sufficiently long strong path-of-sets system has a same-vertex subgraph of maximum degree three in which its first left interface remains polylogarithmically edge-well-linked.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

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    1import Lax17.Degree
    2import Lax17.Linkedness
    3import Lax17.PathOfSets
    4import Mathlib.Data.Nat.Log
    5
    6/-!
    7---
    8title: A low-degree well-linked core
    9type: theorem
    10---
    11A sufficiently long strong path-of-sets system has a same-vertex subgraph of
    12maximum degree three in which its first left interface remains
    13polylogarithmically edge-well-linked.
    14-/
    15
    16namespace Lax17.LowDegreeWellLinkedCore
    17
    18universe u
    19
    20/-- The local strong-path form of the degree-three sparsifier theorem. -/
    21axiom lowDegreeWellLinkedCore :
    22 ∃ cLength logLength cWellLinked logWellLinked : ℕ,
    23 0 < cLength ∧ 0 < logLength ∧
    24 0 < cWellLinked ∧ 0 < logWellLinked ∧
    25 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    26 {G : SimpleGraph V} {length width : ℕ}
    27 (P : Lax17.PathOfSets.StrongSystem G length width),
    28 1 < width →
    29 (∃ half : ℕ, width = 2 * half) →
    30 cLength * (Nat.log 2 width) ^ logLength ≤ length →
    31 ∃ H : SimpleGraph V,
    32 H ≤ G ∧
    33 Lax17.Degree.MaximumAtMost H 3 ∧
    34 Lax17.Linkedness.ScaledEdgeWellLinked H
    35 (P.left ⟨0, P.length_pos⟩) 1
    36 (cWellLinked *
    37 (Nat.log 2 width) ^ logWellLinked)
    38
    39end Lax17.LowDegreeWellLinkedCore
    40
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