Let $G$ be a graph with chromatic number $\aleph_1$. Is it true that there is a colouring of the edges with $\aleph_1$ many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all edge colours?

1True ↔2  ∀ {V : Type u_1} (G : SimpleGraph V),3    G.chromaticCardinal = Cardinal.aleph 1 →4      ∃ EColor,5        ∃ (_ : Cardinal.mk EColor = Cardinal.aleph 1),6          ∃ c_edge,7            ∀ (VColor : Type),8              Cardinal.mk VColor ≤ Cardinal.aleph 0 →9                ∀ (c_vert : V → VColor),10                  ∃ vc,11                    ∀ (ec : EColor), ∃ u v, ∃ (h : G.Adj u v), c_vert u = vc ∧ c_vert v = vc ∧ c_edge ⟨s(u, v), h⟩ = ec