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Lax54.RodlTheorem

Rödl's theorem for induced-subgraph-free graphs

concepts/Lax54/RodlTheorem.lean · lax-54

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    Theorem

    For every finite graph HH and every positive integer EE, 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 such that either G[X]G[X] or its complement has edge density at most 1/E1/E. This is the cleared-denominator finite form of Rödl's theorem cited as Theorem 4.1 in the paper.

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    1import Lax54.GraphDefinitions
    2
    3/-!
    4---
    5title: Rödl's theorem for induced-subgraph-free graphs
    6type: theorem
    7---
    8For every finite graph HH and every positive integer EE, there is a positive
    9integer DD such that every finite induced-HH-free graph GG contains a set
    10XX satisfying ∣V(G)∣≤D∣X∣|V(G)|\leq D|X| and such that either G[X]G[X] or its complement
    11has edge density at most 1/E1/E. This is the cleared-denominator finite form of
    12Rödl's theorem cited as Theorem 4.1 in the paper.
    13-/
    14
    15open scoped SimpleGraph
    16
    17namespace Lax54.RodlTheorem
    18
    19open Lax54.GraphDefinitions
    20
    21universe u v
    22
    23/-- Rödl's theorem, with a reciprocal integer density parameter. -/
    24axiom rodl_theorem :
    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 →
    30 ∃ X : Finset V, Fintype.card V ≤ D * X.card ∧
    31 HasSparseSide (G.induce (X : Set V)) E
    32
    33end Lax54.RodlTheorem
    34
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