Right English wording for "counterexamples to a theorem"

This question is about the right English wording.

I give here what I call "counterexamples to Banach fixed-point theorem". What I do, is that I look to what happen if some hypothesis of the theorem are not fulfilled.

I call this "counterexamples to Banach fixed-point theorem". However, I was told that this is not a proper English wording as a theorem cannot have a counterexample! I'm a French native speaker and it seems that a straight translation of "contre-exemples au théorème du point fixe de Banach" is not good.

What would be the appropriate wording to this situation?

Solution 1:

How about simply "counterexamples related to the Banach fixed point theorem"?

Or if you want to be more precise, "counterexamples to possible strengthenings of the Banach fixed point theorem".

Solution 2:

I am also not a native English speaker, so not sure it this is a good suggestion, but I would use something like Necessity proofs or Necessity demonstrations or Necessity examples (meaning: examples that prove that every assumption of the theorem is necessary).

Solution 3:

A possible English wording is "what happens if we drop the hypothesis of the Banach fixed-point theorem?"

Solution 4:

I have looked at your website before, and I remember being a bit dissatisfied with the terminology "counterexamples"- from a technical point of view. For me, a counter-example is some constructible object that demonstrates the falsehood of some statement. Equivalently, a counter example demonstrates the truth of the negation of the original statement. So for (counter-)example, take the statement "All functions are invertible". That is, for every $f$ there is some $f^{-1}$ such that for all $x$ in the domain of $f$, it is true that $f^{-1}(f(x)) = x$. A counter example to this is the function $f(x) = x^2$. Because, assuming an inverse function $f^{-1}$ we have

$$1 = f^{-1}(f(1)) = f^{-1}(1) = f^{-1}(f(-1)) = -1$$ which is clearly a contradiction. So this is a counter example to the claim.

On the other hand, as one commenter points out, from a "marketing" point of view, "counter-example" sure sounds catchy. But ultimately I find it misleading.

Edit: I realise I didn't really answer your question! In your case, I would say "Counter-examples to variations on the theorem statement" or something along those lines. Or go with the answer above. There are really a lot of ways to say it, but I would certainly avoid "counter examples to the ____ theorem".