TITLE: Reference request: Examples of research on a set with interesting properties which turned out to be the empty set
QUESTION [44 upvotes]: I've seen internet jokes (at least more than 1) between mathematicians like this one here about someone studying a set with interesting properties. And then, after a lot of research (presumably after some years of work), find out such set couldn't be other than the empty set, making the work of years useless (or at least disappointing), I guess.
Is this something that happens commonly? Do you know any real examples of this?
EDIT: I like how someone interpreted this question in the comments as "are there verifiably true examples of this well-known 'urban legend template'?"

REPLY [7 votes]: The odd-order theorem states that every finite group of odd order is solvable, and the proof involves developing a very large theory explaining what the smallest counterexample looks like, and to ultimately deduce that it cannot exist.
The odd-order theorem has been formalised (pdf) in Coq, a computer theorem prover, and the formalisation is to date one of the largest bodies of formalised mathematics. This makes it appealing to AI researchers, who go and train their deep learning networks using the collection of theorems proved in the formalisation, hoping that one day computers will start to be able to compete with humans in the realm of theorem-proving.
I find it amusing that, as a consequence, these networks are being trained to recognise a whole bunch of facts about an object which doesn't exist.