Definitions A tree means a set-theoretic tree, that is a poset $(T,<)$ so that for each $x\in T$, the set $\{y\in T\mid y<x\}$ is well-ordered.
A $\kappa$-Kurepa tree is a tree of height $\kappa$, each level has at most $< \kappa$ many elements and has at least $\kappa^{+}$-many maximal branches.
I know that it is consistent with ZFC the existence of $\kappa$-Kurepa trees, for every $\kappa$. I know it is also consistent the negation of the existence of such trees.
But is it consistent to have $\kappa$-Kurepa trees, for some $\kappa \geq \omega_{1}$, but no $\lambda$-Kurepa trees for $\lambda \neq \kappa$?
Where can I find such independence results for $\kappa$-Kurepa trees?