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Lax54.MaximumDegreeReduction

Maximum-degree form of Rödl's theorem

concepts/Lax54/MaximumDegreeReduction.lean · lax-54

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    Theorem

    Lemma 4.3 of the paper, with denominators cleared. For every finite graph HH and every E>0E>0, there is a positive integer DD such that every finite induced-HH-free graph GG contains a set XX satisfying ∣V(G)∣≤D∣X∣|V(G)|\leq D|X| and, either in G[X]G[X] or in its complement, Edeg⁡(x)<∣X∣E\deg(x)<|X| for every x∈Xx\in X.

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    1import Mathlib.Combinatorics.SimpleGraph.Copy
    2import Lax54.GraphDefinitions
    3
    4/-!
    5---
    6title: Maximum-degree form of Rödl's theorem
    7type: theorem
    8---
    9Lemma 4.3 of the paper, with denominators cleared. For every finite graph HH
    10and every E>0E>0, there is a positive integer DD such that every finite
    11induced-HH-free graph GG contains a set XX satisfying
    12∣V(G)∣≤D∣X∣|V(G)|\leq D|X| and, either in G[X]G[X] or in its complement,
    13Edeg⁡(x)<∣X∣E\deg(x)<|X| for every x∈Xx\in X.
    14-/
    15
    16namespace Lax54.MaximumDegreeReduction
    17
    18open Lax54.GraphDefinitions
    19open scoped SimpleGraph
    20
    21universe u v
    22
    23/-- The maximum-degree form of Rödl's theorem used in Section 4. -/
    24axiom maximum_degree_reduction :
    25 ∀ {W : Type u} [Fintype W] (H : SimpleGraph W) (E : ℕ),
    26 0 < E → ∃ D : ℕ, 0 < D ∧
    27 ∀ {V : Type v} [Fintype V] [DecidableEq V] (G : SimpleGraph V)
    28 [DecidableRel G.Adj],
    29 ¬ H ⊴ G → HasLowDegreeSide G E D
    30
    31end Lax54.MaximumDegreeReduction
    32
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