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Lax17.ExponentTenCrossbarDichotomy

Exponent-ten crossbar dichotomy

concepts/Lax17/ExponentTenCrossbarDichotomy.lean · lax-17

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    Theorem

    The exponent-ten crossbar theorem: three equal terminal sets and two full linkages yield either a width-g2g^2 crossbar or a minor carrying a large strong path-of-sets system.

    Lean source view on GitHub

    1import Mathlib.Data.Nat.Log
    2import Lax17.Crossbar
    3import Lax17.Degree
    4
    5/-!
    6---
    7title: Exponent-ten crossbar dichotomy
    8type: theorem
    9---
    10The exponent-ten crossbar theorem: three equal terminal sets and two full
    11linkages yield either a width-\(g^2\) crossbar or a minor carrying a large
    12strong path-of-sets system.
    13-/
    14
    15namespace Lax17.ExponentTenCrossbarDichotomy
    16
    17universe u
    18
    19/-- The axiom-free exponent-ten crossbar dichotomy. -/
    20axiom exponentTenCrossbarDichotomy :
    21 ∃ c : ℕ, 0 < c ∧
    22 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    23 (G : SimpleGraph V) {A B X : Finset V} {g κ : ℕ},
    24 2 ≤ g →
    25 Lax17.Crossbar.IsPowerOfTwo g →
    26 A.card = κ → B.card = κ → X.card = κ →
    27 Disjoint A B → Disjoint A X → Disjoint B X →
    28 2 ^ 22 * g ^ 10 * Nat.log 2 g ≤ κ →
    29 (∀ x ∈ X, Lax17.Degree.Exactly G x 1) →
    30 Lax17.Paths.VertexLinkage G A B κ →
    31 Lax17.Paths.VertexLinkage G A X κ →
    32 Nonempty
    33 (Lax17.Crossbar.System
    34 G A B X (g ^ 2)) ∨
    35 ∃ length width : ℕ,
    36 g ^ 2 ≤ c * length ∧
    37 g ^ 2 ≤ c * width ∧
    38 Lax17.Crossbar.HasStrongPathOfSetsMinor
    39 G length width
    40
    41end Lax17.ExponentTenCrossbarDichotomy
    42
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