Let $\mathfrak{c}$ be the cardinality of the continuum.  How much Choice, if any, is needed to prove that there are $2^{\mathfrak{c}}$ distinct (mutually nonisomorphic) torsion-free abelian groups of cardinality $\mathfrak{c}$?  This can be proved with AC, but I suspect a much weaker form of Choice, or maybe none at all, is necessary.  A fairly simple argument, not requiring any choice, is that there are at least as many as there are antichains in the power set of $\omega$, and the AC implies there are $2^{\mathfrak{c}}$ such antichains.  Maybe an entirely different argument is possible.