So this is an exercise($4.25$) from Ralf Schindler's book, and I have some trouble with it.
This is the statement:
(Jensen-Kunen) Show that if $\kappa$ is ineffable, then there is no $\kappa$-Kurepa tree.
So naturally after thinking about this for some time, I decided to take a peek at its solution for a hint.
First I found this in Assaf Rinot's webpage. But there he proves this for slim $\kappa$-Kurepa trees. Then I tried to find the main paper where Jensen and Kunen first proved this. But all I found was an unpublished manuscript in Jensen's webpage, which was extremely hard to read for me and even there it seems that the result is proved for slim $\kappa$-Kurepa trees as well, even though they don't mention it.
This is a link to Jensen's webpage. The unpublished paper is called "Some Combinatorial Properties of $L$ and $V$". The result is in page $26$ of the second chapter, it's theorem $9$.
So my question is: Is the above statement correct for general $\kappa$-Kurepa trees? If so, how would one prove it?
Note that whenever $\kappa$ is a strong limit cardinal (and of course ineffable cardinals are strong limit) there is a $\kappa$-Kurepa tree, namely the full binary tree of height $\kappa$, where by $\kappa$-Kurepa tree I mean a tree of height $\kappa$, such that all levels have size $<\kappa$ but the tree has at least $\kappa^+$ branches.
So either Schindler's book has the slimness condition in the definition of a $\kappa$-Kurepa tree or the exercise has a typo, but I'm not familiar with the book so I can't tell. I see that you already found a proof of the fact that if $\kappa$ is ineffable then there is no slim $\kappa$-Kurepa tree, you can also look at theorem 2.6 here, this pdf also seems to cover more of the results contained in Jensen's handwritten notes if you're interested.