Lax Lean Archive

Draft — mutable and not usable as a dependency; its citation marks the draft state.

Lax57.AnticomponentBlockade

Anticomponent or complete blockade

concepts/Lax57/AnticomponentBlockade.lean · lax-57

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Concept map

    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A

    Evidence

    Each proof establishes this claim relative to its assumptions.

    Theorem

    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).

    Lean source view on GitHub

    1import Lax57.GraphDefinitions
    2
    3/-!
    4---
    5title: Anticomponent or complete blockade
    6type: theorem
    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
    Show Proof

    Builds on

    Used by

    none

    From Mathlib

    none

    Community review

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above; your ORCID profile must share a public name.

    0 comments

    Loading discussion…