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It is a theorem of Baumgartner and Laver that iterating Sacks forcings of weakly compact length gives rise to the tree property at $\omega_2$. Natural questions (at least for me) are: do we get stronger tree properties at $\omega_2$ if we start with larger cardinals? In particular, if the length is strongly compact, do we get the strong tree property at $\omega_2$? If the length is supercompact, do we get the super tree property at $\omega_2$? I suspect it is known so I appreciate pointers to references.

For uncountable cardinals $\kappa\leq \lambda$ with $\kappa$ being regular, we say $F\subset \bigcup_{u\in [\lambda]^{<\kappa}} 2^u$ is a $(\kappa,\lambda)$-tree if it is closed under taking restrictions, and for each $u\in [\lambda]^{<\kappa}$, $|F_u:=\{f\in F: dom(f)=u\}|<\kappa$. A function $d: \lambda\to 2$ is a cofinal branch if for all $u\in [\lambda]^{<\kappa}$, $d\restriction u\in F_u$. Given $\bar{f} = \langle f_u\in F: dom(f_u)=u\rangle$ (called a level sequence), we say $d: \lambda\to 2$ is an $\bar{f}$-ineffable branch if $\{u\in [\lambda]^{<\kappa}: d\restriction u = f_u\}$ is stationary in $[\lambda]^{<\kappa}$.

We say $\kappa$ has the strong tree property if for any $\lambda\geq \kappa$, any $(\kappa, \lambda)$-tree has a cofinal branch.

We say $\kappa$ has the super tree property if for any $\lambda\geq \kappa$, any $(\kappa, \lambda)$-tree $F$ and any level sequence $\bar{f}$, there exists an $\bar{f}$-ineffable branch.

Edit: I just realized the answer in the case where the length is strongly compact is yes: the countable support iteration of Sacks forcings of strongly compact length forces the Semistationary Reflection Principle (equivalent to the statement that all stationary preserving forcings are semiproper and in fact stronger statement, i.e. the Baire version of Rado's conjecture, holds), along with $\neg CH$, implies the strong tree property holds at $\omega_2$ by Torres-Perez and Wu (https://www.researchgate.net/publication/306394279_Strong_Chang's_Conjecture_Semi-Stationary_Reflection_the_Strong_Tree_Property_and_two-cardinal_square_principles). The question remains for the case where the length is supercompact.

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  • $\begingroup$ What you've called a $(\kappa, \lambda)$-tree seems to also be known as a $(\kappa, \lambda)$-mess (Definition 2.1 of sciencedirect.com/science/article/pii/0003484373900144/… ). Moreover, in that same paper, it's stated that for inaccessible $\lambda$, $\lambda$ is strongly-compact, iff, every "mess" is solvable (equiv. has a cofinal branch.) (Removed previous comment to fix errors and tone.) $\endgroup$ – Not Mike Jan 8 '18 at 21:30
  • $\begingroup$ Thought I'd add, that definition 4.1 (again form that same paper) and subsequent item 4.2 might be of interest. $\endgroup$ – Not Mike Jan 8 '18 at 21:42
  • $\begingroup$ It seems the answer is positive for the strongly compact case. $\endgroup$ – Jing Zhang Jan 9 '18 at 2:13
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    $\begingroup$ @NotMike: HA! I KNEW IT!!!! A couple of years ago I was talking with Yair Hayut about these things, and I said that the term "mess" fits them much better. Not just because it's messy, but also it works better with the terminology from the proofs about the equivalence of the compactness theorem for logic and completeness and BPI (over ZF). It's good to finally know that this is a thing. But anyway, nowadays the term "tree" is more common, unfortunately. $\endgroup$ – Asaf Karagila Jan 9 '18 at 2:42
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    $\begingroup$ @AssfKaragila sigh.. I mean... I guess it's kinda like a "tree".. if you restrict yourself to only those $f\in F$ with $dom(f)\in C$ where C is some appropriate club set. (Was going to make a joke about hitting a tree with a club but couldn't put it together..) $\endgroup$ – Not Mike Jan 9 '18 at 3:46
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The answer to the supercompact case is yes. More specifically, in the forcing extension obtained by iterating Sacks forcing of supercompact length, the super tree property at $\omega_2$ holds. This follows from the following: 1) Countable support iteration of Sacks forcing satisfies $\omega_1$-approximation property. This was essentially shown in Baumgartner and Laver's classical paper about iterated Sacks forcing 2) Theorem 5.4 in Weiss' paper: Combinatorial Essence of Supercompactness (https://www.sciencedirect.com/science/article/pii/S0168007211001904).

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