Adding Additional Coauthors
While I have never had more than one coauthor, I have spoken to others who, like myself, have been in a situation where there were two coauthors who had worked together for a long time on a project and had become stuck. Here are a few scenarios that were described to me anonymously. In all three scenarios, two co-authors have spent many years on a project together with significant numbers of propositions proven before becoming stuck on a step towards their overarching goal:
1) One of the coauthors asks a colleague from his department a question and the colleague answers the question. It is not just a simple matter of pointing to a reference (which would get an acknowledgement). In fact the colleague has written up a detailed multipage proof. At this point there are three things the coauthors considered: (a) offering co-authorship on the whole paper , (b) asking him to write an appendix to their paper, (c) asking him to publish his theorem somewhere so they could cite it. The third colleague turned down co-authorship on the whole paper in part because it would have taken him time to follow the entire paper in a field outside of his own and, as a pure mathematician, putting his name on the paper would indicate a certain level of understanding the whole thing. Ultimately he chose (b) rather than (c) thinking that his result wasn't interesting enough to stand on its own. In fact I know two scenarios in which this is how things went. In terms of the mathematics, (b) is the ideal choice. In terms of promotion, a junior mathematician might benefit more from a separate paper (c). Certainly the paper can mention direct applications of the results will appear in another paper, thus boosting its chances for publication even if it doesn't stand on its own as interesting. I also know of a case where the senior mathematician really didn't have any more time than to outline a proof and so he asked only to be listed in the acknowledgements as specifically suggesting the proof of the specific theorem. In fact, at the referee stage, it is not uncommon for a referee to find an error and then to provide a suggestion for a key step that is needed (I often do this), and then the referee is only acknowledged anonymously.
2) One of the coauthors is traveling to visit an old collaborator. He asks his coauthor if he can mention their difficulties to his old trusted collaborator. They decide that if the old trusted collaborator has an idea, then he can be offered co-authorship on the entire paper. They are not concerned that the trusted collaborator will steal their work and complete it on his own. In the end the coauthor with the old trusted collaborator works out the problem and the three publish a jointly coauthored paper together.
3) One of the coauthors notices there is a mathematician who has recently published work that might be helpful to complete their project. The coauthor invites this mathematician to present at a seminar and, after the seminar, asks the visitor if he thinks that his work can help finish their project. They speak for hours. The other mathematician goes home, writes a paper completing the project and posts it on the arxiv as sole authored work. The coauthor who spoke to him feels like an idiot for not emphasizing that this was meant to be a joint project or clarifying what would happen. The other coauthor is furious with his coauthor as well as the visitor. They quickly finalize a paper (since they had been texing all along) which includes all their results before the discussion and post it on the arxiv within a week. The visitor does not view himself as a thief because he viewed the discussion as growing out of his own work and doesn't see the reframing of the question as a serious effort on the part of the other two mathematicians. He adds a citation to their paper begrudgingly, when confronted. Ultimately the original pair get a quietly published paper while the visitor's paper is published in a top journal. Without the arxiv, the original pair might not have even gotten a publication. The visitor might have just submitted his paper for publication and none would be the wiser.
In the three stories, there is an issue of trust when asking a new person for help. Clearly the new person must be agreed upon by the first two collaborators and the new person must be informed that this is their project.
If a project is truly stuck, there is also the possibility of publishing your work with conjectures as to how to finish and giving it away to the world at large. One might say this is how Hamilton worked, publishing papers on Ricci flow, openly discussing how it might be used to solve the Poincare Conjecture and what difficulties were arising to complete the program. When Perelman was able to complete the project, he cited Hamilton's work and Hamilton achieved a lot of recognition from the Geometric Analysis community for starting the program that ultimately lead to the proof of the Poincare Conjecture. In fact many great papers were published over the years with progress towards the Poincare Conjecture. One does not need to completely finish a problem alone or with one's collaborators. There are times when it is best to publish what you've done and let people know about it.