Erdős problem 1175 · problems.science

Let $\kappa$ be an uncountable cardinal. Must there exist a cardinal $\lambda$ such that every graph with chromatic number $\lambda$ contains a triangle-free subgraph with chromatic number $\kappa$ ?

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FormalConjectures/ErdosProblems/

1175.lean

Retained formal statement1 of 5

Let $\kappa$ be an uncountable cardinal. Must there exist a cardinal $\lambda$ such that every graph with chromatic number $\lambda$ contains a triangle-free subgraph with chromatic number $\kappa$ ?

Shelah proved that a negative answer is consistent when $\kappa = \lambda = \aleph_1$ (see erdos_1175.variants.shelah_consistency).

FormalConjectures/ErdosProblems/1175.leanErdos1175.erdos_11756 linesExact file

1True ↔2  ∀ (κ : Cardinal.{u_1}),3    Cardinal.aleph0 < κ →4      ∃ μ,5        ∀ (V : Type u_1) (G : SimpleGraph V),6          G.chromaticCardinal = μ → ∃ H, H.coe.CliqueFree 3 ∧ H.coe.chromaticCardinal = κ

OpenStatement only, no proof

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