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A simple existence criterion for normal spanning trees in infinite graphs

Halin proved in 1978 that there exists a normal spanning tree in every connected graph $G$ that satisfies the following two conditions: (i) $G$ contains no subdivision of a `fat' $K_{\aleph_0}$, one in which every edge has been replaced by uncountably many parallel edges; and (ii) $G$ has no $K_{\aleph_0}$ subgraph. We show that the second condition is unnecessary.