Erdős Problem #1067 — Chromatic Subgraphs with Infinite Connectivity

No. The following assertion is false: every simple graph $G$ whose chromatic cardinal is $\aleph_1$ contains a subgraph $H$ whose chromatic cardinal is $\aleph_1$ and which is infinitely connected. Equivalently, for every type of vertices $V$ and every simple graph $G$ on $V$, if $G$ has chromatic cardinal $\aleph_1$, then there exists a subgraph $H$ of $G$ such that $H$ has chromatic cardinal $\aleph_1$ and is infinitely connected.