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Cut-matching expansion theorem

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

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    Natural Language Statement

    Theorem

    On every finite even vertex set, a logarithmic number of perfect matching rounds can be chosen across successive bisections so that the union of the matching-edge instances has constant expansion. This is the existential form of the cut-matching game used in the grid-minor proof; algorithmic running-time claims are intentionally omitted.

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    1import Mathlib.Data.Nat.Log
    2import Lax17.Expansion
    3
    4/-!
    5---
    6title: Cut-matching expansion theorem
    7type: theorem
    8---
    9On every finite even vertex set, a logarithmic number of perfect
    10matching rounds can be chosen across successive bisections so that the union
    11of the matching-edge instances has constant expansion. This is the
    12existential form of the cut-matching game used in the grid-minor proof;
    13algorithmic running-time claims are intentionally omitted.
    14-/
    15
    16namespace Lax17.CutMatchingTheorem
    17
    18universe u
    19
    20/-- A cut-matching transcript of \(O(\log n)\) rounds with half-expansion. -/
    21axiom logarithmicCutMatchingExpansion :
    22 ∃ c : ℕ, 0 < c ∧
    23 ∀ (V : Type u) [Fintype V] [DecidableEq V],
    24 2 ≤ Fintype.card V →
    25 (∃ half : ℕ, Fintype.card V = 2 * half) →
    26 ∃ T : Lax17.Expansion.CutMatchingTranscript V,
    27 T.length ≤ c * Nat.log 2 (Fintype.card V) ∧
    28 T.IsHalfEdgeExpander
    29
    30end Lax17.CutMatchingTheorem
    31
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