  We consider a transitive relation on the power set of $\omega_1$ and show if
there is a maximal element with respect to this relation then there is a Kurepa
tree with no Aronszajn subtree. We also show that if there is a maximal subset
of $\omega_1$, then there are Kurepa trees which are not club isomorphic. These
maximal subsets of $\omega_1$ exist in many known models that are obtained from
the constructible universe without large cardinal assumptions. For instance,
whenever $\alpha_0 \in \omega_1$ and $X \subset \omega_1$ are such that
$\omega_1^{\textsc{L}[X \cap \alpha_0]} = \omega_1, \omega_2^{\textsc{L}[X]} =
\omega_2$ and $\textsc{V}$ is a semiproper forcing extension of $\textsc{L}[X]$
then $X$ is maximal in $\textsc{V}$.
