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In a recent question I've asked if a claim I've make could be true, a user answered with a counterexample and so I've accepted his answer.

Now I formulated some additional hypothesis under which I think my claim is true, but I still can't prove it. Clearly it's not right to reopen my old question, but is it right to ask a new one with this improvement?

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It is OK, if you attribute appropriately and you are not a hog about it.

Although you can update a question, unless done very carefully, it can invalidate previous answers. One of the times I've gotten very mad is when I post an answer to a question showing that some aspect needs more consideration, and the poster deletes the question along with my answer! I now work hard on ignoring that user's questions, because I do not want my efforts dismissed that way. So respect the work people have put in to help by acknowledging it, even if they delete it afterwards but you still found it helpful.

If you do make version 2.0 (or even 1.1) of the question, have each question refer to the other so that interested parties can follow the development trail. If you start posting version 1.2 or 1.3 however, take that as a sign that you need to do more thinking on the subject. In particular, one thing I think we should see more of in these situations are comments along the lines of "what should I be asking here?". You then get the advice of professional question-askers on how and what to ask.

Gerhard "Purely An Amateur Questioner Answerer" Paseman, 2017.03.27.

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    $\begingroup$ Of course no valuable answer should be deleted via question deletion. -- Though on the other hand I doubt that it is good practice to unnecessarily clutter the site with questions which are not well-thought and answers which merely point that out. $\endgroup$ – Stefan Kohl Mar 27 '17 at 17:04
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    $\begingroup$ Indeed. That is why I usually comment first, to prod the user into formulating a better question. However, with one person (I believe it was one using two different ids) comments were not enough, and my subsequent answers were deleted along with the questions. Indeed clutter not MathOverflow with such attempts, but I think we need more examples of how the attempts are made and corrected. Gerhard "It Is A Learning Process" Paseman, 2017.03.27. $\endgroup$ – Gerhard Paseman Mar 27 '17 at 17:26

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