If $f$ and $g$ are two functions, define $f \sim g$ if they differ only finitely often on their common domain.

The following property of a large cardinal arose from a problem in model theory. I am interested in its strength. Say that a cardinal $\kappa$ has the weak tree property if the following holds:

Suppose $(b_\alpha: \alpha < \kappa)$ is a sequence such that:

• Each $b_\alpha \in 2^\alpha$,
• For each $\alpha < \beta < \kappa$, $b_\alpha \sim b_\beta$.

Then there is some $b \in 2^\kappa$ such that for all $\alpha < \kappa$, $b \sim b_\alpha$.

The following summarizes what I know about the weak tree property:

• If $\kappa$ has the tree property then it has the weak tree property. (A sequence $(b_\alpha)$ such as above can also be viewed as defining an Aronszajn tree.)
• $\aleph_1$ does not have the weak tree property. (One of the standard constructions of an $\aleph_1$ Aronszajn tree yields this.)
• If $\kappa$ has countable cofinality then $\kappa$ has the weak tree property.

I am interested in knowing more. For example, might $\aleph_2$ not have the weak tree property? (It is possible that $\aleph_2$ has the weak tree property since it might have the full tree property.)

Any thoughts would be appreciated.