# On a weak tree property for inaccessible cardinals

Suppose that $\kappa$ is inaccessible and consider a tree of height $\kappa$ whose levels have size strictly below some cardinal $\gamma < \kappa$. Does this type of tree always have a $\kappa$-branch?

The answer is known to be positive when $\gamma=\omega$, i.e., when the levels are finite. On the other hand, if the levels are only known to have size strictly below $\kappa$ itself, then the answer is negative in general since the existence of a $\kappa$-branch implies that $\kappa$ is weakly compact.

I would be interested in knowing about references/proofs on whether this weak tree property holds for every inaccessible or if in fact it constitutes a large cardinal axiom whose strength is somewhere between inaccessibility and weak compactness. Ideally, I would expect a proof of a positive answer in ZFC (one can assume that the inaccessible is given).

Theorem Suppose that $\kappa = cf(\kappa) > \gamma$, and $(T, <_{T})$ is a $\kappa$-tree each of whose levels has cardinality less than $\gamma$. Then $(T, <_{T})$ has a cofinal branch.
The proof uses the Pressing Down Lemma (Fodor's theorem) and the Pigeonhole Principle with the bound $\gamma$ to find the branch.