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Lax57.ErdosHajnalP5

Erdős–Hajnal theorem for the five-vertex path

concepts/Lax57/ErdosHajnalP5.lean · lax-57

proven

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    Theorem

    The five-vertex path P5P_5 has the Erdős–Hajnal property: there is a positive integer qq such that every finite graph GG with no induced copy of P5P_5 satisfies

    ∣V(G)∣≤max⁡{α(G),ω(G)}q.|V(G)| \leq \max\{\alpha(G),\omega(G)\}^{q}.

    Lean source view on GitHub

    1import Lax57.GraphDefinitions
    2
    3/-!
    4---
    5title: Erdős–Hajnal theorem for the five-vertex path
    6type: theorem
    7---
    8The five-vertex path P5P_5 has the Erdős–Hajnal property: there is a positive
    9integer qq such that every finite graph GG with no induced copy of P5P_5
    10satisfies
    11
    12
    13∣V(G)∣≤max⁡{α(G),ω(G)}q.|V(G)| \leq \max\{\alpha(G),\omega(G)\}^{q}.
    14
    15-/
    16
    17namespace Lax57.ErdosHajnalP5
    18
    19open Lax57.GraphDefinitions
    20
    21universe u
    22
    23/-- The Erdős–Hajnal conjecture for the five-vertex path. -/
    24axiom erdos_hajnal_P5 :
    25 ∃ q : ℕ, 0 < q ∧
    26 ∀ {V : Type u} [Fintype V] (G : SimpleGraph V),
    27 IsP5Free G → Fintype.card V ≤ homogeneousNumber G ^ q
    28
    29end Lax57.ErdosHajnalP5
    30
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