There is a tree property in set-theory and a tree property in model theory. I want the former, not the later.
Definition The tree property holds at some cardinal $\kappa$ if every tree of height $\kappa$ with levels of size less than $\kappa$ has a cofinal branch.
Counterexamples to the tree property at $\kappa$ are called $\kappa$-Aronszajn trees.
In a few instances, I heard people talking about the tree property as a "compactness property". One of the justifications was the following theorem.
Theorem An inaccessible cardinal has the tree property if and only if it is weakly compact.
Since compactness is a very useful tool in model-theory, one would expect that in the absence of compactness, e.g. in infinitary logics, the tree property to be some sort of replacement. Looking around I did not find anything.
So, my question is:
Do you know any usage of the (set-theoretic) tree property in model theory? Even better, do you know any model-theoretic techniques that use the tree property? By technique, I mean something that can be used in various settings, not in only one particular construction.