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I am submitting a mathematics paper to a journal. This paper contains a section where I claim that some results made by some authors in different published articles are false. I give a counterexample to support my claim and also show where their proofs are flawed.

Should I oppose these authors as reviewers when I submit the manuscript? In this case what should I write in the “reason” section?

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    Why do you want to oppose? Why not keep them as reviewers? Won't they be happy finding the mistake? – Coder Aug 4 '17 at 21:28
  • Well normally, I should keep them as reviewer. But I don't know how they will react when criticizing their works. – user144542 Aug 4 '17 at 21:39
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    I would think the authors in question would belong to the set of ideal reviewers for your paper. Additionally, I assume that you criticized their work in a respectful manner. (You did respectfully critique their work, right?) – Mad Jack Aug 4 '17 at 21:54
  • As @MadJack alludes to, as long as you were respectful in your critique, those authors would easily be the ideal reviewers for your paper, as they are intimately aware of the problem which you're working on. – deckeresq Aug 4 '17 at 21:55
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    In an ideal world, you would want these people as your referees. In this one, maybe ask around or google to see if they are known to be difficult or respond poorly to criticism. There are some people who are like this. – AJK Aug 4 '17 at 21:59
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If possible, it is good form to contact these people a few days before you submit so they can "check" your counterexample. Phrase your counterexample in a constructive manner v.g. avoid saying it is false but rather suggest their proof is incomplete.

If these people are reasonable, they will double check, confirm that the counterexample is valid, thank you for it, and maybe cite your own work. If they are not reasonable you have still been gracious.

People don't like to be criticized in public - nobody likes it. Avoid "springing a trap" that will get these people in a bad mood if they learn of your counterexample in a journal or as referee of the paper.

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    Right. Use polite, non-aggressive language such as "I don't see why X follows from Y" or, "I seem to have found counter-example X. Perhaps I'm not understanding something. Can you help me understand the situation?" – paul garrett Aug 4 '17 at 22:13
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    ... after all, maybe you're mistaken!?! :) – paul garrett Aug 4 '17 at 22:45
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    @user144542, I understand. Probably you are correct. Still, one must always admit the possibility of error, and to behave in a way that directly acknowledges/disclaims the possibility of error will always be greeted more happily than the pose of "being sure I'm right". Even if you do feel sure, others have checked it, etc. Just take the modest position, if you'd ask my advice. – paul garrett Aug 5 '17 at 0:53
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    "Phrase your counterexample in a constructive manner v.g. avoid saying it is false but rather suggest their proof is incomplete." An incomplete proof is something completely different to a false theorem. If somebody tells me my proof is incomplete, I'll try to fix the gap. That would be a total waste of time if there's a counterexample to the theorem's statement, since that means that there can be no possible fix to the proof. So, sure, be diplomatic and use phrases like "seems to be incorrect" rather than "is wrong" or "is false", but don't sugar-coat so much that you're lying. – David Richerby Aug 5 '17 at 13:24
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    @DavidRicherby Of course with so little details the spirit is more important than the letter of my answer, but your turn of phrase is probably better than mine. – user67075 Aug 5 '17 at 15:38
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Your concern is that some potential reviewers have a conflict of interest. In my opinion, it is the editor's responsibility to manage conflicts of interest. You should inform the editor of the facts in your cover letter, and leave it at that.

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You shouldn't oppose reviewers unless you have reasonable belief that they'll be biased against your paper. That might be because they're biased against you, your particular sub-field or the techniques you've used; or, in cases such as this one, if they're very sensitive to any kind of criticism. In this case, we can be pretty sure that they're not biased against your sub-field (they wrote papers in it themselves!) or really against your techniques (that doesn't seem to be a thing in mathematics, but more likely in experimental sciences).

Simply being proven objectively wrong shouldn't cause significant bias for most mathematicians. Of course, it's embarrassing to have published something wrong. However, it seems that the error wasn't central to the original paper, or they'd have tried to write an actual proof without the supposedly redundant assumption. It seems to have been more of an aside, so the level of embarrassment we're talking about is "Oops, I made a boo-boo" and nowhere near "Noooooo! My career is destroyed! I must bury this!"

The authors of the claim you showed to be wrong are likely to be particularly good reviewers, since they understand the topic very well. I'd expect the natural choice of reviewers would include one of those people and one outsider.

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