Lax Lean Archive

Rödl's theorem for induced-subgraph-free graphs

Lax54.RodlTheorem · concepts/Lax54/RodlTheorem.lean · lax-54

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Lemma

    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.

    Concept map
    2 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    In the paper

    • page 10 of this submission's paper
    • page 2 of the paper of lax-57, Erdős–Hajnal for the five-vertex path

    Lean source view on GitHub

    1import Lax54.GraphDefinitions
    2
    3/-!
    4---
    5title: Rödl's theorem for induced-subgraph-free graphs
    6type: lemma
    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
    Show Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…