Recall that a tree on $\omega\_1$ is called Suslin if every chain and antichain are countable. If every level is countable and there are no cofinal branches, then it is called Aronszajn (in particular Suslin is Aronszajn). Moreover, an Aronszajn tree is called special if it is the countable union of antichains.


So special Aronszajn trees and Suslin trees represent some sort of endpoints on the spectrum of Aronszajn trees: either every antichain is countable, or the whole tree is a countable union of antichains.


Given an Aronszajn tree, we can always specialize it using a ccc forcing; and given a Suslin tree it is a ccc forcing by itself, and forcing with it will add a cofinal branch (which is an uncountable chain).


Therefore, given a Suslin tree, we have a choice of how we want to violate its Suslinity: either violate its Aronszajn-ness, or specialize it.



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> Can we make a Suslin tree into a non-special Aronszajn tree, while destroying its Suslinity?
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If the answer is positive, what is the "best" way of doing so? (Either consistently or provably, from the existence of a Suslin tree.) 


Can we do it with a ccc forcing, or with a $\sigma$-closed forcing, or with just a proper forcing? Do the properties of this forcing somehow depend on the tree (e.g. if the tree is rigid, or homogeneous, etc.)?


(As a minor bonus question, is it consistent that all Aronszajn trees are Suslin or special, but Suslin trees exist?)




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Just a remark on triviality, when I say "tree" I always mean that any point has at least two successors, and that it has successors on every level of the tree. I am also going to assume the tree is Hausdorff, in the sense that at limit levels the points are always determined by their predecessors.



Destroying Suslin, nothing special
Chapter IX of Proper and Improper Forcing addresses this issue.


Shelah proves that Souslin's Hypothesis does not imply every Aronszajn tree is special, and he does this by investigating weak notions of specialness that are still incompatible with Souslinity. He shows that there are forcings that "specialize" Aronszajn trees in the weak sense, and that they can be iterated while still preserving at least one non-special Aronszajn tree.


Corollary 4.8 spells out more details:


He starts with a Souslin tree $T^\*$ and preserves the fact that it is a non-special Aronszajn tree while also specializing all A-trees in a weak sense. He uses an $\aleph\_1$-free iteration of proper forcing instead of countable support, but it is not clear this is necessary.


The concept of $(T^\*, S)$-preserving (Definition 4.5) seems to be the property of forcing notions that ensures the tree $T^\*$ never gets fully specialized. 


The entire chapter is pretty technical, but it contains many interesting ideas that could and should be developed further.


