  Diestel and Leader have characterised connected graphs that admit a normal
spanning tree via two classes of forbidden minors. One class are Halin's
$(\aleph_0,\aleph_1)$-graphs: bipartite graphs with bipartition
$(\mathbb{N},B)$ such that $B$ is uncountable and every vertex of $B$ has
infinite degree.
  Our main result is that under Martin's Axiom and the failure of the Continuum
Hypothesis, the class of forbidden $(\aleph_0,\aleph_1)$-graphs in Diestel and
Leader's result can be replaced by one single instance of such a graph.
  Under CH, however, the class of $(\aleph_0,\aleph_1)$-graphs contains
minor-incomparable elements, namely graphs of binary type, and
$\mathcal{U}$-indivisible graphs. Assuming CH, Diestel and Leader asked whether
every $(\aleph_0,\aleph_1)$-graph has an $(\aleph_0,\aleph_1)$-minor that is
either indivisible or of binary type, and whether any two
$\mathcal{U}$-indivisible graphs are necessarily minors of each other. For both
questions, we construct examples showing that the answer is in the negative.

Combinatorics;Logic