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Lax5.AlmostLinearNC

Monadically dependent classes have almost linear neighborhood complexity

concepts/Lax5/AlmostLinearNC.lean · lax-5

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    Theorem

    Every monadically dependent graph class has almost linear neighborhood complexity: for every ε > 0 there is a constant c such that every member G and every nonempty vertex subset A satisfy |{N(v) ∩ A : v ∈ V(G)}| ≤ c · |A|^(1+ε).

    This is Theorem 2 of Dreier, Mählmann, McCarty, Pilipczuk, Toruńczyk, Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes (2026).

    Lean source view on GitHub

    1import Lax5.MonadicDependence
    2import Lax12.NeighborhoodComplexity
    3
    4/-!
    5---
    6title: Monadically dependent classes have almost linear neighborhood complexity
    7type: theorem
    8---
    9Every monadically dependent graph class has almost linear neighborhood
    10complexity: for every ε > 0 there is a constant *c* such that every
    11member *G* and every nonempty vertex subset *A* satisfy
    12|{N(v) ∩ A : v ∈ V(G)}| ≤ *c* · |A|^(1+ε).
    13
    14This is Theorem 2 of Dreier, Mählmann, McCarty, Pilipczuk, Toruńczyk,
    15*Neighborhood Complexity and Radius-1 Merge-Width in Monadically
    16Dependent Graph Classes* (2026).
    17
    18# Formalization notes
    19
    20The hypothesis is the transduction-based definition of monadic
    21dependence, the subject of this submission; the conclusion is the
    22predicate `HasAlmostLinearNC` of the *Sparsity Lectures* submission
    23(Lax12), where neighborhood complexity is defined and endorsed. Stating
    24the theorem over that predicate is what makes it directly comparable to
    25the nowhere dense counting statement there: the same bound, under a
    26strictly weaker hypothesis.
    27-/
    28
    29namespace Lax5.AlmostLinearNC
    30
    31open Lax12.GraphClasses Lax12.NeighborhoodComplexity Lax5.MonadicDependence
    32
    33/-- Monadically dependent graph classes have almost linear neighborhood
    34complexity. -/
    35axiom hasAlmostLinearNC_of_monadicallyDependent
    36 (C : GraphClass) (h : MonadicallyDependent C) :
    37 HasAlmostLinearNC C
    38
    39end Lax5.AlmostLinearNC
    40
    Show Proof

    Formalization notes

    The hypothesis is the transduction-based definition of monadic dependence, the subject of this submission; the conclusion is the predicate HasAlmostLinearNCHasAlmostLinearNC of the Sparsity Lectures submission (Lax12), where neighborhood complexity is defined and endorsed. Stating the theorem over that predicate is what makes it directly comparable to the nowhere dense counting statement there: the same bound, under a strictly weaker hypothesis.

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