A question of Erdős and Hajnal [ErHa68b], who proved that for every finite $k$ there is a graph with chromatic number $\aleph_1$ and $\aleph_k$ vertices where each subgraph on less than $\aleph_k$ vertices has chromatic number $\leq \aleph_0$.

1True ↔2  ∀ (ω : Ordinal.{u}),3    ∃ V G,4      Cardinal.mk V = Cardinal.aleph (ω + 1) ∧5        G.chromaticCardinal = Cardinal.aleph 1 ∧6          ∀ (H : G.Subgraph), Cardinal.mk ↑H.verts = Cardinal.aleph ω → H.coe.chromaticCardinal ≤ Cardinal.aleph0