Necessary use of large cardinals in mathematics There are some statements, whose consistency (or the consistency of their negation) require the existence of large cardinals (in the sense that if the statement (or its negation) is consistent, then it is consistent that there are some large cardinals). I know many of such examples inside set theory.

Question. What are examples of  statements in  mathematics (other than set theory), where it is known we need large cardinals to prove their consistency 
  (or the consistency of their negation)? 

For each example, giving what kind of large cardinals are sufficient and  what kind of large cardinals are necessary is appreciated. Please also give references.
 A: The dual of an abelian group $A$ is defined to be the group $\text{Hom}(A,\mathbb Z)$ of homomorphisms to the infinite cyclic group. As usual with such dualities, there's a canonical homomorphism from $A$ to its double dual
$$
A\to A^{**}:a\mapsto(h\mapsto h(a)).
$$
If this is an isomorphism, $A$ is said to be reflexive. 
Question: Are all free abelian groups reflexive?
Answer: Yes if and only if there are no measurable cardinals.
A: Recall that the character of a point in a topological space is the smallest cardinality of a local base for that point. (So, for example, "first countable" = "every point has character $\aleph_0$".)

Question: Is there a compact Hausdorff space, containing more than one point, in which any two different points have different character?

Steve Watson found an answer to this question [S. Watson, "Using prediction principles to construct ordered continua," Pacific Journal of Mathematics 186, pp. 251-256 (link)] by showing that
Theorem: There is such a space if and only if there is a cardinal $\kappa$ and a set of cardinals $E \subseteq \kappa$ such that $\diamondsuit_\kappa(E)$ holds.
Watson points out in his paper that this condition is implied by $V=L$ plus the existence of a Mahlo cardinal, and it implies the existence of a weakly inaccessible cardinal. Thus

the consistency strength of a positive answer to the above question lies somewhere between an inaccessible cardinal and a Mahlo cardinal.

I do not know whether better bounds for this statement about $\diamondsuit_\kappa(E)$ have been found in the last 34 years. If you know of any improvements to the bounds Watson gives, please feel free to edit this post.
Finally, Watson shows from ZFC alone that there is an infinite $\sigma$-compact Hausdorff space in which any two different points have different character. Thus removing "compact" from the question makes it much easier, and removes the need for any large cardinals.
A: It was a traditional question of descriptive set theory (a question which can be formulated in the language of second order arithmetic) whether all projective sets  are Lebesgue measurable. This remained an open problem for many decades, and for a good reason: it turned out that the statement is independent even of the full ZFC set theory (see Solovay 1970). Only by postulating the existence of some extremely large cardinals (so-called Woodin cardinals) can the hypothesis that all projective sets are Lebesgue measurable be proved (this was achieved as a consequence of their work on so-called projective determinacy by Woodin, Martin and Steel; see Woodin 1988; Martin & Steel 1988, 1989).
