Asking this question here as it was suggested in one of the comments from MO.
I recently refereed a paper that I returned to the author(s) for revision. The thrust of their argument relied on a claim whose justification I felt was lacking. I dutifully raised the issue in my report and, in addition, I corrected another portion of their proof. I also obtained partial results on this problem from another paper (I initially raised the problem they purport to prove).
The author(s) have yet to revise their work and, in the interim, I came up with a justification for their claim. I now have a proof of this result (which is important for another paper I'm working on) and would like to publish it (at least obtain some level of attributionfor the correct proof).
What are the ethics/options here? Do I need to provide them with the correct proof? May I submit the result as my own? If so, must I wait until their revision before I submit my own paper? Should I disentangle myself from any further involvement as the referee?
EDIT: I’d like to add a little context to one of the points made in a response below.
The conjecture they purport to prove is one that I made in a previous paper which included partial results on the conjecture. Thus, I have thought about the problem and the “ownership” of the problem belongs to me.
However, it is clear that I wouldn’t have come up with the proof without their submission.
It seems to me that a co-authorship of some sort would be the best for all parties, or I simply pass along the proof to the author(s).