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What if this happens:

Say I contacted a researcher R in my field about particular question I had. They answered the question in a private email to me. Their sketch of the argument was about 5 sentences long, and it can be turned into a rigorous proof. I really like the result. However, R has been unresponsive for 2 months and has not replied to any of my subsequent emails where I make various offers and seek their advice.

Would it be unethical to include their argument as a small part of a publication I'm preparing, assuming I give credit to R? What constitutes proper credit in this case? A sentence in the introduction when I first mention the result? A sentence preceding the proof? In Acknowledgements?

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In most places a citation (ack) would be enough. For a citation, list it as a "private communication". That is normally enough. You avoid plagiarism because you properly credit the idea.

There may be a few places, though I can't name them, where it requires specific permission to do this. It isn't needed in the US, but seek local guidance elsewhere.

Most likely you will be paraphrasing the communication rather than quoting it. Quoting a communication in full may have different rules, even if it is only a few sentences. To do more, you need permission.

One way to get permission, or at least acceptance, is to send the person a copy of your finished paper, saying that it is about to be submitted and asking for any advice on it. If they have issues with how you have presented their ideas they have a chance to correct them.

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    I think permission should be explicitly granted, regardless of whether it is "required" in some formal sense. Give a call to the person in question and ask them explicitly. I don't like the "private communication" citations. They are essentially useless: impossible to verify, and there is no actual document being referenced. My advice would be to mention them in the acknowledgments, in the appropriate part of the introduction, add another acknowledgement in a sentence immediately preceding the statement of the result, and of course in the heading of the result itself. Papers are not ... – Andrés E. Caicedo Jun 16 '19 at 0:07
  • ... usually read linearly, and it is better to be a bit redundant to ensure the authorship of the result is made transparent to anyone looking at the paper. – Andrés E. Caicedo Jun 16 '19 at 0:08
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  1. Write the paper, attributing the person in the way you feel most suitable. I would write that the person in question provided an outline/idea of the proof of proposition this and that, and include something to this effect in the acknowledgements, close to the result, and maybe in the introduction if the result is a major one for the paper.

  2. Send an email the person with the attached draft, writing that you finished a preprint/paper using their idea, is the attribution fine or (if earned) do they want coauthorship? Maybe also write that you'll be submitting the paper to a preprint server/journal/conference in two weeks (or whatever), so you would prefer an answer by then. If not, you'll submit the paper as is.

  3. Give the email a descriptive but short title. If you are fairly unknown to the person, try to not write a title which looks like spam or crankery.

  4. If they do not react and you are still uncertain about the attribution, you may try calling or visiting or some other means of communication.

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