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I am currently in the process of trying to publish a mathematics paper. A draft of this paper had been posted on arxiv, which contained an interesting conjecture.

In the intervening time, this conjecture has been refuted by another group of authors. I am not sure how I should edit my paper to reflect this.

One option that is out of the question is to simply delete the conjecture from the paper. The problem with this is that the latter paper has cited this conjecture as its motivation. If I removed the conjecture, I would be pulling the legs out from the subsequent paper.

What I am leaning toward is to state something along the lines of "In an earlier draft, we had made the following conjecture:" and then include some discussion and citations about how the conjecture has subsequently been refuted.

Note that most of my paper is unaffected by this refutation, and in fact I still believe that the conjecture could hold for the special case I need

Is there a better way of handling this situation?

  • AFAIK, arxiv stores every version as separate paper. So your original work is more or less cast in stone and cited already, you can only follow-up – aaaaaa Nov 28 '18 at 20:20
  • Where have they published? If it's an easily-editable format (e.g. arXiv and not print) you could always contact them and ask them to cite a specific version, because you're editing the paper and don't want their citation to become difficult to trace. – Nic Hartley Nov 29 '18 at 19:48
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If you can prove or at least make a convincing case for the conjecture in the special case you need, I'd do the following:

  1. Describe the conjecture in the special case
  2. Prove or make the convincing case in the special case
  3. Write "in a previous draft, we expected this conjecture to hold generally. However, it has subsequently been shown ..."

If you can't or don't want to argue for the conjecture in your special case, then your formulation works as well.

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When you believe, that your special case still holds, you can keep the conjecture and mention that you believe it holds for your special case and that it does not hold in the general case with a citation of the other paper.

We conjecture X for all cases in which Y holds. Note that X does not hold in the general case, as shown by Name [42].

So the other paper even improves on the revision of your paper and you can acknowledge this by adding this information and the citation. It would be way more complicated when the conjecture would be totally wrong. But when it is correct for a special case, both papers are useful.

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One option that is out of the question is to simply delete the conjecture from the paper. The problem with this is that the latter paper has cited this conjecture as its motivation. If I removed the conjecture, I would be pulling the legs out from the subsequent paper.

This isn't pulling the legs out, since the subsequent paper cites a published arxiv draft. Indeed, Tyszka published 152 drafts (of one work) over almost six years and any version can be cited.

What I am leaning toward is to state something along the lines of "In an earlier draft, we had made the following conjecture:" and then include some discussion and citations about how the conjecture has subsequently been refuted.

That's certainly a nice touch.

I still believe that the conjecture could hold for the special case I need

Prove this result, at least partially.

  • “Prove this result.” Presumably the OP has already thought seriously about this and can’t yet see how, otherwise they would have included it as a theorem not a conjecture. – PLL Nov 28 '18 at 13:09
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    @PLL The OP believes the conjecture holds for a special case, even though the conjecture has been disproved in the general case. So, I presume the OP has a significant basis for their belief. At the very least, they can surely prove part of the special case, thereby reducing the leap of faith required to accept it. – user2768 Nov 28 '18 at 13:12
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    I believe it COULD hold for the special case (i.e. it is still a relevant, interesting, and open, conjecture). I have no evidence that it DOES hold. – David Harris Nov 28 '18 at 14:04
  • @DavidHarris Without any evidence, I personally wouldn't call it a conjecture. But, the definition of conjecture is rather vague, so your usage is correct. I tried searching for a mathematician's definition: I couldn't find one. (I did stumble upon Guido's book of conjectures, which is quite nice.) – user2768 Nov 28 '18 at 14:21
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    @user2768: I’d say mathematicians typically use conjecture for a statement we have some heuristic evidence of — e.g. various particular cases we know hold, or perhaps some numerical evidence — but no proof, and usually not even anything we’d call a partial proof, at the point where we first present it as a conjecture (since if we had a partial proof, we’d usually then isolate what was missing to complete it and present that statement as the conjecture instead). – PLL Nov 28 '18 at 14:49

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