Lax Lean Archive

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

Lax57.SparseHouseAcceleration

Sparse-house acceleration

concepts/Lax57/SparseHouseAcceleration.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

    For some integer d≥2d\geq 2, a house-free graph that is 1/R21/R^2-sparse on a set SS, where R≥64R\geq 64, either becomes 1/R2d1/R^{2d}-sparse on a subset of size at least ∣S∣/R32d3|S|/R^{32d^3}, or has a complete or anticomplete RR-blockade whose blocks have size at least ∣S∣/R36d3|S|/R^{36d^3}. This is a reciprocal-square form of Lemma 7.2 of Nguyen, Scott, and Seymour.

    Lean source view on GitHub

    1import Lax57.GraphDefinitions
    2
    3/-!
    4---
    5title: Sparse-house acceleration
    6type: theorem
    7---
    8For some integer d≥2d\geq 2, a house-free graph that is 1/R21/R^2-sparse on a
    9set SS, where R≥64R\geq 64, either becomes 1/R2d1/R^{2d}-sparse on a subset of
    10size at least ∣S∣/R32d3|S|/R^{32d^3}, or has a complete or anticomplete RR-blockade
    11whose blocks have size at least ∣S∣/R36d3|S|/R^{36d^3}. This is a reciprocal-square
    12form of Lemma 7.2 of Nguyen, Scott, and Seymour.
    13-/
    14
    15namespace Lax57.SparseHouseAcceleration
    16
    17open Lax57.GraphDefinitions
    18
    19universe u
    20
    21/-- The sparse-house acceleration step, in reciprocal-square form. -/
    22axiom sparse_house_acceleration :
    23 ∃ d : ℕ, 2 ≤ d ∧
    24 ∀ R : ℕ, 64 ≤ R →
    25 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    26 (G : SimpleGraph V) [DecidableRel G.Adj] (S : Finset V),
    27 IsHouseFree G → ESparse G (R ^ 2) S →
    28 (∃ T : Finset V, T ⊆ S ∧
    29 S.card ≤ R ^ (32 * d ^ 3) * T.card ∧
    30 ESparse G (R ^ (2 * d)) T) ∨
    31 (∃ B : Blockade (V := V) R,
    32 B.IsInside S ∧ B.IsUniform G ∧
    33 ∀ i : Fin R,
    34 S.card ≤ R ^ (36 * d ^ 3) * (B.block i).card)
    35
    36end Lax57.SparseHouseAcceleration
    37
    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…