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Lax57.SparseHouseTrichotomy

The sparse-house trichotomy

concepts/Lax57/SparseHouseTrichotomy.lean · lax-57

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    Theorem

    This is a denominator-cleared form of Lemma 7.1 of Nguyen, Scott, and Seymour. At scale QQ, a sparse house-free graph either becomes polynomially sparser on a polynomial fraction of its vertices, contains a long complete blockade, or admits an anticomplete peel. In the last alternative, the set YY omits at most a 3/Q3/Q fraction of the current vertex set.

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    1import Lax57.GraphDefinitions
    2
    3/-!
    4---
    5title: The sparse-house trichotomy
    6type: theorem
    7---
    8This is a denominator-cleared form of Lemma 7.1 of Nguyen, Scott, and
    9Seymour. At scale QQ, a sparse house-free graph either becomes polynomially
    10sparser on a polynomial fraction of its vertices, contains a long complete
    11blockade, or admits an anticomplete peel. In the last alternative, the set
    12YY omits at most a 3/Q3/Q fraction of the current vertex set.
    13-/
    14
    15namespace Lax57.SparseHouseTrichotomy
    16
    17open Lax57.GraphDefinitions
    18
    19universe u
    20
    21/-- Sparse refinement, a complete blockade, or an anticomplete peel. -/
    22axiom sparse_house_trichotomy :
    23 ∃ d : ℕ, 40 ≤ d ∧
    24 ∀ Q : ℕ, 8 ≤ Q →
    25 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    26 (G : SimpleGraph V) [DecidableRel G.Adj] (S : Finset V),
    27 IsHouseFree G → ESparse G Q S →
    28 ( (∃ T : Finset V, T ⊆ S ∧
    29 S.card ≤ Q ^ (30 * d ^ 3) * T.card ∧
    30 ESparse G (Q ^ (2 * d)) T) ∨
    31 (∃ B : Blockade (V := V) Q,
    32 B.IsInside S ∧ B.IsComplete G ∧
    33 ∀ i : Fin Q,
    34 S.card ≤ Q ^ (33 * d ^ 3) * (B.block i).card) ∨
    35 (∃ X Y : Finset V,
    36 X ⊆ S ∧ Y ⊆ S ∧ Disjoint X Y ∧
    37 (∀ x ∈ X, ∀ y ∈ Y, ¬ G.Adj x y) ∧
    38 S.card ≤ Q ^ (33 * d ^ 3) * X.card ∧
    39 Q * (S.card - Y.card) ≤ 3 * S.card) )
    40
    41end Lax57.SparseHouseTrichotomy
    42
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