Publishing someone's argument 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?
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.