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erdos_918.variants.all_subgraphs.parts.ii

Specification

Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?

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Lean 4 Statement 
theorem erdos_918.variants.all_subgraphs.parts.ii :
    answer(sorry) ↔ ∀ (ω : Ordinal),
      ∃ (V : Type u) (G : SimpleGraph V), #V = ℵ_ (ω + 1) ∧ G.chromaticCardinal = ℵ₁ ∧
      ∀ (H : G.Subgraph) (_ : #H.verts = ℵ_ ω), H.coe.chromaticCardinal ≤ ℵ₀
ID: ErdosProblems__918__erdos_918.variants.all_subgraphs.parts.ii
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