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).



On a weak tree property for inaccessible cardinals
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.


