# Does $\mathsf{MA}^+(\sigma-{\rm closed})$ imply there are no Kurepa Trees?

The question in the title is somewhat self contained but let me make some definitions and remarks to clarify.

Recall that $$\mathsf{MA}^+(\sigma-{\rm closed})$$ is the statement that if $$\mathbb P$$ is a $$\sigma$$-closed poset, $$\langle D_\alpha \; |\; \alpha < \omega_1\rangle$$ is a sequence of dense subsets of $$\mathbb P$$ and $$\dot{S}$$ is a name for a stationary set then there is a filter $$G \subseteq \mathbb P$$ so that $$G \cap D_\alpha \neq \emptyset$$ for all $$\alpha < \omega_1$$ and $$\{\alpha \; | \; \exists p \in G \; p \Vdash \check{\alpha} \in \dot{S}\}$$ is stationary.

A Kurepa tree is a tree of height $$\omega_1$$, countable levels and $$\geq \omega_2$$ branches.

My question is simply if $$\mathsf{MA}^+(\sigma - {\rm closed})$$ implies there are no Kurepa trees.

I have no intuition about this, as on the one hand I don't see a way that $$\sigma$$-closed forcing suffices to show there are no Kurepa trees and on the other hand I don't see a way to force $$\mathsf{MA}^+(\sigma-{\rm closed})$$ without killing all Kurepa trees. Specifically:

It's well known that Silver collapsing an inaccessible to $$\omega_2$$ implies that there are no Kurepa trees so in natural models of $$MA^+(\sigma-{\rm closed})$$ obtained by iterating all $$\sigma$$-closed forcing notions below some sufficiently large inaccessible (supercompact?) $$\kappa$$ there are no Kurepa Trees.

Meanwhile, the proof that $$\mathsf{PFA}$$ implies there are no Kurepa trees involves specializing an Aronszajn tree and such specialization forcings are not in general $$\sigma$$-closed.

Thanks!

Yes, it implies no Kurepa trees. First, note that the forcing axiom you consider implies the Weak Reflection Principle, which in turn implies (a strong form of ) Chang's Conjecture. Both of those facts are covered in the Foreman-Magidor-Shelah paper on Martin's Maximum. And Chang's Conjecture implies there are no Kurepa trees. I believe the proof of the latter appears in Foreman's chapter in the Handbook of Set Theory, but here is a sketch: suppose $$T$$ was a Kurepa tree on $$\omega_1$$; let $$\langle b_i \ : \ i < \omega_2 \rangle$$ be a 1-1 list of cofinal branches of $$T$$. By Chang's Conjecture, there is an $$X \prec (H_{\omega_3},\in, \vec{b})$$ such that $$X \cap \omega_2$$ has ordertype $$\omega_1$$, but $$X \cap \omega_1 \in \omega_1$$. Consider the collection $$\{ b_i \ : \ i \in X \cap \omega_2 \}$$. For any distinct $$b_i$$, $$b_j$$ in that collection, since $$i \ne j$$ and both indices are in $$X$$, it follows that $$b_i$$ and $$b_j$$ diverge at some level below $$X \cap \omega_1$$. This implies that level $$X \cap \omega_1$$ of $$T$$ has size $$\omega_1$$, contradicting that it is a thin tree.