Erdős Problem 1068

Reference: erdosproblems.com/1068

theorem Erdos1068.erdos_1068 : sorry ↔ ∀ (V : Type) (G : SimpleGraph V), ↑G.chromaticNumber = Cardinal.aleph 1 → ∃ (s : Set V), s.Countable ∧ (SimpleGraph.induce s G).InfinitelyConnected

Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely connected?