TITLE: Result that follows from ZFC and not ZF but are strictly weaker than choice
QUESTION [7 upvotes]: A number of results that people use that require the axiom of choice (i.e. do not follow from ZF alone) are known to actually imply the axiom of choice. Therefore, one might naturally wonder whether there are results that require choice to prove yet which, on the contrary, do not imply choice.
The most natural answer to this question is countable choice, or $\kappa$-choice for cardinals $\kappa$.
Therefore, I wonder, are there other (i.e. not equivalent to any of the above) examples of results with this property? What about which are strictly stronger or weaker than all the cardinal-choice axioms (but not equivalent to ZF or ZFC)? In particular, I am interested in results occurring naturally in algebra, geometry, etc. And whether or not there are, is there a deeper reason why results that use choice tend to imply it?

REPLY [3 votes]: Here's one that I think a lot of people don't realize: the existence of free algebras for all (possibly infinitary) algebraic theories.  In his paper "Words, free algebras, and coequalizers" Andreas Blass proved that (assuming the consistency of a proper class of compact cardinals) it is impossible to prove this in ZF.  (I don't know for sure that it is strictly weaker than AC, but it seems to me extremely unlikely to be equivalent to AC.)  I think this is especially interesting because it doesn't look like any sort of choice principle itself.