# Logical relationships between weakenings of AC

What are the known logical implications between weak choice principles like $DC_\kappa$", the ultrafilter 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), andevery 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?

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Just a couple of minor quibbles here; first, your statement of the ultrafilter theorem is not quite correct because a principal ultrafilter cannot be extended to a nonprincipal ultrafilter, and second, you probably meant to ask about "how either of these consequences of AC relates to DC$_\kappa$," not to uniformization. – Trevor Wilson Jul 17 '12 at 21:20
Also, to answer a very small part of your question, in Solovay's paper "The independence of DC from AD" a natural model of AD is considered in which: 1) every relation on $\mathbb{R}$ admits a uniformization, but 2) there is an $\omega$-sequence of nonempty subsets of $\mathcal{P}(\mathbb{R})$ with no choice function, and 3) there is an $\omega_1$-sequence of nonempty subsets of $\mathbb{R}$ with no choice function (because of AD.) – Trevor Wilson Jul 17 '12 at 21:31