Short research articles

[a bit too long for a comment]

I understand from the question that the aim is to find a research project based on the search for a counterexample. By construction, this will mean showing that some existing paper in the literature is mistaken. That is typically not a productive way to start a project in a new field, simply because (a) if the author of that paper is clueless then there is not much gained in showing them wrong by finding a counterexample,$^\ast$ while (b) if the author is an expert you are facing an uphill battle if you are just entering a field.

Typically, a more productive way to enter a field is to try to generalize/extend work of others, basically by exploring corners of the field they left untouched (or didn't bother to explore). You may find that this leads you to uncover an error/oversight in the paper you started from, but that would then be a byproduct of your research and not the primary motivation.

_{$^\ast$ many questions here on MathOverflow can be readily dismissed by finding a counterexample, but that rarely becomes something worthy of a publication}

For counterexamples in analysis, a good start would be "Counterexamples in Analysis" by Gelbaum and Olmsted, there's even a Dover edition.