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Lax57.SemisparseBlockade

Polynomial semisparse blockades for the house

concepts/Lax57/SemisparseBlockade.lean · lax-57

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    Theorem

    This is the denominator-cleared form of Lemma 6.2 of Nguyen, Scott, and Seymour. A sufficiently large house-free graph contains EE disjoint polynomially large blocks, and every pair of blocks is either complete or has edge density at most E−dE^{-d}.

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    1import Lax57.GraphDefinitions
    2
    3/-!
    4---
    5title: Polynomial semisparse blockades for the house
    6type: theorem
    7---
    8This is the denominator-cleared form of Lemma 6.2 of Nguyen, Scott, and
    9Seymour. A sufficiently large house-free graph contains EE disjoint
    10polynomially large blocks, and every pair of blocks is either complete or
    11has edge density at most E−dE^{-d}.
    12-/
    13
    14namespace Lax57.SemisparseBlockade
    15
    16open Lax57.GraphDefinitions
    17
    18universe u
    19
    20/-- Polynomial semisparse blockades in house-free graphs. -/
    21axiom semisparse_house_blockade :
    22 ∃ d : ℕ, 40 ≤ d ∧
    23 ∀ E : ℕ, 2 ≤ E →
    24 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    25 (G : SimpleGraph V) [DecidableRel G.Adj],
    26 IsHouseFree G → E ^ (10 * d ^ 2) ≤ Fintype.card V →
    27 ∃ B : Blockade (V := V) E,
    28 B.IsSemisparse G (E ^ d) ∧
    29 B.HasWidthLoss (E ^ (10 * d ^ 2))
    30
    31end Lax57.SemisparseBlockade
    32
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