On a weak tree property for inaccessible cardinals

Question

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).

Answer 1

The following theorem of Kurepa, proved in his thesis of 1935, seems to address your question.

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.

A good reference on the general area is A. Kanamori, The Higher Infinite, which contains the proof of the above theorem. You might also wish to consult J. Cummings, M. Foreman, The tree property, Advances in Mathematics, 131, 1998, 1-32.