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 $\leq \aleph_0$.

1∀ (k : ℕ),2  0 < k →3    ∃ V G,4      Cardinal.mk V = Cardinal.aleph ↑k ∧5        G.chromaticCardinal = Cardinal.aleph 1 ∧6          ∀ (W : Set V),7            Cardinal.mk ↑W < Cardinal.aleph ↑k → (SimpleGraph.induce W G).chromaticCardinal ≤ Cardinal.aleph0