Lax Lean Archive

Monadically dependent classes exclude large φ-patterns

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

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    Natural Language Statement

    Corollary

    For every monadically dependent graph class 𝒞 and formula φ(x̄, z), there is some t ∈ ℕ such that no graph G ∈ 𝒞 contains a φ-pattern of order t × t.

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

    Each proof establishes this claim relative to its assumptions.

    In the paper

    • page 23 of this submission's paper

    Lean source view on GitHub

    1import Lax185574.PhiPatterns
    2import Lax185574.MonadicDependence
    3
    4/-!
    5---
    6title: Monadically dependent classes exclude large φ-patterns
    7type: corollary
    8---
    9For every monadically dependent graph class 𝒞 and formula φ(x̄, z),
    10there is some t ∈ ℕ such that no graph G ∈ 𝒞 contains a φ-pattern of
    11order t × t.
    12
    13# Formalization notes
    14
    15The paper allows infinite graphs in this part. Here the graphs are
    16those of a `GraphClass`, which are finite, as in every use.
    17
    18This is the case ψ(y, z) = E(y, z) or ψ(y, z) = ¬E(y, z) of the
    19exclusion of (φ, ψ)-patterns. A single t excludes both kinds of
    20φ-patterns. The formula is a formula of the language of graphs with
    21free variables as in `PhiPsiPatterns`.
    22-/
    23
    24namespace Lax185574.ForbiddenPhiPatterns
    25
    26open FirstOrder FirstOrder.Language
    27open Lax185574.Graphs Lax185574.MonadicDependence Lax185574.PhiPatterns
    28
    29/-- **Forbidden `φ`-patterns.** No graph of a monadically dependent
    30class contains a `φ`-pattern of order `t × t`, for some `t` depending on
    31the class and `φ`. -/
    32axiom forbidden_phi_patterns (C : GraphClass) (hC : MonadicallyDependent C)
    33 {p : ℕ} (φ : Language.graph.Formula (Fin p ⊕ Fin 1)) :
    34 ∃ t : ℕ, ∀ G ∈ C, ¬ HasPhiPattern G.toSimpleGraph φ t t
    35
    36end Lax185574.ForbiddenPhiPatterns
    37
    Show Proof
    Formalization notes

    The paper allows infinite graphs in this part. Here the graphs are those of a GraphClassGraphClass, which are finite, as in every use.

    This is the case ψ(y, z) = E(y, z) or ψ(y, z) = ¬E(y, z) of the exclusion of (φ, ψ)-patterns. A single t excludes both kinds of φ-patterns. The formula is a formula of the language of graphs with free variables as in PhiPsiPatternsPhiPsiPatterns.

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