For a strong limit cardinal $\kappa$ the notion of *$\kappa$-Kurepa tree* is trivial: the full binary tree is a $\kappa$-Kurepa tree. Accordingly, we consider the following strengthening:


A *slim $\kappa$-Kurepa tree* is a tree $T$ of height $\kappa$ such that for every infinite $\alpha < \kappa$ the $\alpha$-th level of $T$ has cardinality $\left| \alpha \right|$, and $T$ has more than $\kappa$ many branches.


If $\kappa$ is a strong limit cardinal of countable cofinality, it's easy to construct a slim $\kappa$-Kurepa tree. On the other hand, if $\kappa$ is measurable (or just ineffable) then there is no slim $\kappa$-Kurepa tree. If $\kappa$ is inaccessible, then my understanding from comments [here](http://blog.assafrinot.com/?p=1964) is that there is a $\mathord{<}\kappa$-closed forcing to create a slim $\kappa$-Kurepa tree (but this destroys measurability.) What about the uncountable cofinality singular case?



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> If $\kappa$ is a singular strong limit cardinal of uncountable cofinality, can there exist a slim $\kappa$-Kurepa tree?
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EDIT: This turned out to be fairly easy; see my answer below. However, I would like to know where I can find this result proved (or at least mentioned) in print. So I will accept the first answer that tells me this.



Slim Kurepa tree at a singular strong limit cardinal of uncountable cofinality
The following is proved by Erdos-Hajnal-Milner in ``[On sets of almost disjoint subsets of a set](http://bolyai.hu/~p_erdos/1968-08.pdf). Acta Math. Acad. Sci. Hungar 19 1968 209–218'', from which the required result follows


**Theorem.** ssume $\aleph\_0 < cf(\kappa) < \kappa$ and $\forall \theta< \kappa, \theta^{cf(\kappa)} < \kappa.$ Let $F \subseteq P(\kappa)$
be such that $\{\alpha < \kappa: |F \restriction \alpha| \leq \alpha \}$
is stationary. Then $|F| \leq \kappa.$


