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Lax54.ErdosHajnalC5

Erdős–Hajnal theorem for the five-cycle

concepts/Lax54/ErdosHajnalC5.lean · lax-54

proven

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    Theorem

    The five-cycle has the Erdős–Hajnal property. Equivalently, there is a positive integer qq such that every finite graph GG with no induced five-cycle satisfies ∣V(G)∣≤max⁡{α(G),ω(G)}q|V(G)|\leq \max\{\alpha(G),\omega(G)\}^q.

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    1import Lax54.GraphDefinitions
    2
    3/-!
    4---
    5title: Erdős–Hajnal theorem for the five-cycle
    6type: theorem
    7---
    8The five-cycle has the Erdős–Hajnal property. Equivalently, there is a
    9positive integer qq such that every finite graph GG with no induced
    10five-cycle satisfies
    11∣V(G)∣≤max⁡{α(G),ω(G)}q|V(G)|\leq \max\{\alpha(G),\omega(G)\}^q.
    12-/
    13
    14namespace Lax54.ErdosHajnalC5
    15
    16open Lax54.GraphDefinitions
    17
    18universe u
    19
    20/-- The Erdős–Hajnal property for the five-cycle, in integer-exponent form. -/
    21axiom erdos_hajnal_C5 :
    22 ∃ q : ℕ, 0 < q ∧
    23 ∀ {V : Type u} [Fintype V] (G : SimpleGraph V),
    24 IsC5Free G → Fintype.card V ≤ homogeneousNumber G ^ q
    25
    26end Lax54.ErdosHajnalC5
    27
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