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.