Asaf Karagila
I don't have much choice...

Talks. Giving talks. We usually don't give talks about past research. Talks are meant to present recent research, things you've just finished, that you're finishing right now, that you've found out!

So often times, it seems, it is very tempting to talk about theorems that you haven't finished writing their proofs in full. Usaully, we put "work in progress" to indicate that this is something not fully verified, not fully vetted (at the very least by ourselves).

Natasha Dobrinen presented at the 2017 Arctic Set Theory meeting a "Theorem*", namely a theorem whose proof is 2/3 written, and has fairly solid structure laid out. It turns out that this notation is also due to Vera Fischer going as far back as 2012. But the two are independent on this discovery, which might as well be a folklore notation.

In this work we build on the Dobrinen–Fischer ideas to extend the notation.

1. Theorem: this is a statement whose proof has been written down, preferably it was read by a few people and received a modicum of approval.
2. Theorem*: this is the Dobrinen–Fischer notation, this is a theorem whose proof has been laid down, and written to a reasonable point, but not entirely.
3. Theorem**: this is a statement whose proof has some structure, but only a little bit of content. We have a general strategy for attacking the problem, we have proved some preliminary lemmas and propositions, but the bulk of the proof is still not written down in a statisfactory manner.
4. Theorem***: this is a statement whose proof does not exist, but a proof strategy does. We have a good idea as to how to prove the statement, and we have evidence as why this should work. But we don't have an exact outline or lemmas.
5. Theorem****: this is just a conjecture, a hypothesis. We have a statement, and we have a strong feeling as to why it is true, but nothing more.
6. Theorem*****: this is an open question, whose answer is wide open.

Thus, we obtain the following easy corollary.

Corollary. A five stars theorem is an open question.

I like this system, for talks, anyway. I wouldn't want to see it catching on in the literature itself.

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