Lax Lean Archive

Polynomial neighborhood complexity from bounded VC-dimension

Lax185574.NeighborhoodComplexityBound · concepts/Lax185574/NeighborhoodComplexityBound.lean · lax-185574

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    Natural Language Statement

    Lemma

    If G has VC-dimension d then

    π_G(k) ≤ (k + 1)^d for all k ∈ ℕ.

    This follows from the Sauer–Shelah lemma.

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

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    In the paper

    • page 13 of this submission's paper

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    1import Lax185574.VCDimension
    2
    3/-!
    4---
    5title: Polynomial neighborhood complexity from bounded VC-dimension
    6type: lemma
    7---
    8If G has VC-dimension d then
    9
    10π_G(k) ≤ (k + 1)^d for all k ∈ ℕ.
    11
    12This follows from the Sauer–Shelah lemma.
    13
    14# Formalization notes
    15
    16The hypothesis is VC-dimension at most d, which is equivalent: a graph
    17of VC-dimension d' ≤ d satisfies the bound for d', hence for d. The
    18graph may be infinite. Its VC-dimension is then an extended natural
    19number, and the hypothesis excludes the value ⊤.
    20-/
    21
    22namespace Lax185574.NeighborhoodComplexityBound
    23
    24open Lax185574.VCDimension
    25
    26universe u
    27
    28/-- **Neighborhood complexity bound.** A graph of VC-dimension at most `d` has
    29neighborhood complexity `π_G(k) ≤ (k + 1)^d`. -/
    30axiom neighborhoodComplexity_le {V : Type u} (G : SimpleGraph V) (d : ℕ)
    31 (hG : vcDim G ≤ d) (k : ℕ) : neighborhoodComplexity G k ≤ (k + 1) ^ d
    32
    33end Lax185574.NeighborhoodComplexityBound
    34
    Show Proof
    Formalization notes

    The hypothesis is VC-dimension at most d, which is equivalent: a graph of VC-dimension d' ≤ d satisfies the bound for d', hence for d. The graph may be infinite. Its VC-dimension is then an extended natural number, and the hypothesis excludes the value ⊤.

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