What are the known logical implications between weak choice principles like
$DC_\kappa$", theultrafilter theorem for sets of size $\kappa$" (by which I mean every filter over a set A of size $\kappa$ can be extended to a non-principle ultrafilter), and ``every relation $R$ over $A$ can be uniformized by a function $f$ with the same domain as $R$"?
Jech's book on the Axiom of Choice gives nice proofs that ``uniformization for $A$" and "the ultrafilter theorem for sets of size $|A|$ logically independent, i.e. neither implies the other. My concern is how either of these consequences of AC relate to uniformization.
I assume $DC_\kappa$ does not imply uniformization of sets of size larger than $\kappa$ but I have yet to see any proof of this fact.
Are the three principles logically independent of one another on a global scale? How do they relate on a more local level?