Lax Lean Archive

Anticomponent or complete blockade

Lax57.AnticomponentBlockade · concepts/Lax57/AnticomponentBlockade.lean · lax-57

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    Natural Language Statement

    Lemma

    This denominator-cleared form of Lemma 4.1 groups the connected components of the complement. Either one anticonnected component has size at least ∣T∣/Q2|T|/Q^2, or there are QQ pairwise complete groups, each of size at least ∣T∣/(4Q3)|T|/(4Q^3).

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

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    In the paper

    • page 6 of this submission's paper

    Lean source view on GitHub

    1import Lax57.GraphDefinitions
    2
    3/-!
    4---
    5title: Anticomponent or complete blockade
    6type: lemma
    7---
    8This denominator-cleared form of Lemma 4.1 groups the connected components
    9of the complement. Either one anticonnected component has size at least
    10∣T∣/Q2|T|/Q^2, or there are QQ pairwise complete groups, each of size at least
    11∣T∣/(4Q3)|T|/(4Q^3).
    12-/
    13
    14namespace Lax57.AnticomponentBlockade
    15
    16open Lax57.GraphDefinitions
    17
    18universe u
    19
    20/-- A large anticonnected component or a long complete blockade. -/
    21axiom anticomponent_or_complete_blockade :
    22 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    23 (G : SimpleGraph V) [DecidableRel G.Adj] (T : Finset V) (Q : ℕ),
    24 2 ≤ Q →
    25 ( (∃ J : Finset V, J ⊆ T ∧
    26 (Gᶜ.induce (J : Set V)).Connected ∧
    27 T.card ≤ Q ^ 2 * J.card) ∨
    28 (∃ C : Blockade (V := V) Q,
    29 C.IsInside T ∧ C.IsComplete G ∧
    30 ∀ i : Fin Q,
    31 T.card ≤ 4 * Q ^ 3 * (C.block i).card) )
    32
    33end Lax57.AnticomponentBlockade
    34
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