TITLE: I am looking for class of math problems which are provable in ZF if and only if they are provable in ZFC
QUESTION [11 upvotes]: I know that P vs NP and Riemannian hypothesis are of this class but could not find any article on that. I would also appriciate links or books on related theme. My question is: what are some other problems of the same class and how one proves such property? I am by the way undergrad of math.

REPLY [4 votes]: I wrote a short note about exactly this.

Asaf Karagila, Absolutely Choiceless Proofs, arXiv:1402.3048 (2014)

The idea is, as the others have remarked, to use absoluteness, but not just for questions on number theory. Instead we focus on a set theoretic absoluteness.
We say that a quantifier is ordinal bounded if it quantifies about sets of finite tuples of ordinals (using the usual coding this is the same as quantifying over sets of ordinals, but it does make the statement a bit more natural).

Theorem. Suppose that $\varphi(x)$ is a formula that is upwards absolute between $L$ and $V$, then $\forall^{\sf Ord}x\,\varphi(x)$ is provable from $\sf ZF$ if and only if it is provable from $\sf ZFC$.

That is a somewhat trivial observation, but it formalises the idea that is used in quite a few places. For example, many questions in transfinite Ramsey theory are framed as "given a partition of $[\alpha]^n$, is there a homogeneous set of such and such properties?", and since the focus is partitions of ordinals, this is exactly an example for a statement that is quantified with ordinal bounded quantifiers.
(I should warn that the note was intended to be fleshed out into a paper, but that never happened since I moved my focus to other problems.)