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Lax57.BlockadeThinning

Simultaneous thinning of a semisparse blockade

concepts/Lax57/BlockadeThinning.lean · lax-57

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    Theorem

    This finite greedy lemma is the sampling-and-cleaning step in Claim 7.1.1. Subblocks are selected in order so that every weakly sparse pair has a controlled edge count. Removing vertices of excessive cross-degree then gives vertexwise sparsity in both directions while retaining at least half of each sample.

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    1import Lax57.GraphDefinitions
    2
    3/-!
    4---
    5title: Simultaneous thinning of a semisparse blockade
    6type: theorem
    7---
    8This finite greedy lemma is the sampling-and-cleaning step in Claim 7.1.1.
    9Subblocks are selected in order so that every weakly sparse pair has a
    10controlled edge count. Removing vertices of excessive cross-degree then
    11gives vertexwise sparsity in both directions while retaining at least half
    12of each sample.
    13-/
    14
    15namespace Lax57.BlockadeThinning
    16
    17open Lax57.GraphDefinitions
    18
    19universe u
    20
    21/-- Thin every block simultaneously and convert weak pair density to
    22vertexwise sparsity. -/
    23axiom semisparse_blockade_thinning :
    24 ∀ {V : Type u} [Fintype V] [DecidableEq V]
    25 (G : SimpleGraph V) [DecidableRel G.Adj]
    26 (k P E t : ℕ) (B : Blockade (V := V) k),
    27 2 ≤ k → 2 * k ≤ t → 128 * k ^ 3 * E ≤ P →
    28 (∀ i, 2 * t ≤ (B.block i).card) → B.IsSemisparse G P →
    29 ∃ C : Blockade (V := V) k,
    30 (∀ i, C.block i ⊆ B.block i) ∧
    31 (∀ i, t ≤ 2 * (C.block i).card ∧ (C.block i).card ≤ t) ∧
    32 (∀ {i j}, i ≠ j →
    33 ((∀ x ∈ C.block i, ∀ y ∈ C.block j, G.Adj x y) ∨
    34 (ESparseTo G E (C.block i) (C.block j) ∧
    35 ESparseTo G E (C.block j) (C.block i))))
    36
    37end Lax57.BlockadeThinning
    38
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