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?
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1$\begingroup$ You didn't say so, but for Kurepa trees probably one intends the height is regular. Otherwise, if $\lambda$ is a strong limit, then $2^{<\lambda}$ is a $\lambda$-tree with $2^\lambda$ many branches. $\endgroup$– Joel David HamkinsCommented Jun 18 at 3:08
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It seems to me that the following construction will work.
Start in $V$ with a Kurepa tree on $\omega_1$, and assume that there are a proper class of inaccessible cardinals above. By cutting off, if necessary, we may assume that there are no inaccessible limits of inaccessibles.
Now perform the Easton-support class iteration that successively makes the inaccessible cardinals the successor cardinals, starting with the first one becoming $\omega_3$, with conditions of size $\omega_1$.
The whole forcing is $\leq\omega_1$-closed, and so will preserve the Kurepa tree you have on $\omega_1$.
But since you made the first inaccessible $\omega_3$, it follows by Silver's argument that there is no Kurepa tree on $\omega_2$. If there were such a tree, it would be added by a piece of the first Levy collapse, and then that tree would gain no new branches in the rest of the collapse, since the forcing is $\leq\omega_1$-closed, and if it had a new branch, then by extending $\omega_1$ times you'd get a level of the tree that was $2^{\omega_1}$, which would mean it wasn't a $\omega_2$-tree.
And similarly for the higher cardinals. Since there were no inaccessible limits of inaccessibles, every regular cardinal $\lambda\geq \omega_3$ was originally inaccessible, and then the collapse making the next inaccessible $\lambda^+$ will prevent there from being any $\lambda$-Kurepa tree.
So altogether, we have a model $V[G]$ that has a Kurepa tree at $\omega_1$, but not at any higher regular cardinal.
answered Jun 18 at 3:32
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1$\begingroup$ One can modify the construction to achieve the Kurepa tree not at $\omega_1$ but at another desired $\aleph_\alpha$. Simply kill all the Kurepa trees down low. Force the one you want at the desired cardinal. And then do the iteration above to kill them all above. In this way, you have it at $\omega_{17}$ or whatever you want, not just $\omega_1$. $\endgroup$ Commented Jun 18 at 4:06
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$\begingroup$ Could you suggest me some bibliography that contains such results for $\kappa$-Kurepa trees? $\endgroup$ Commented Jun 19 at 4:25
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1$\begingroup$ I think the basic fact is in Jech's book. I don't know any other sources, sorry, but perhaps someone can post some helpful items. $\endgroup$ Commented Jun 19 at 4:52