A function from to is called *even* if for all , . We might call it *even about the point* if, for all , .

Conversely, we can call a function *strongly non-even* if for all , , .

Finding strongly non-even functions is easy, as any injective function provides a trivial example. We can make things harder for ourselves by considering only functions from to . But now, it is just as easy to show that there are no strongly non-even functions.

Therefore, let’s make the following definition: Let a function be *non-even of order* if, for all , . Thus, a strongly non-even function is non-even of order , and a function being non-even of order implies that it’s non-even of order for all .

In this paper, the set theorists Peter Komjáth and Saharon Shelah proved:

The existence of a non-even function of order 1 is equivalent to the Continuum Hypothesis (i.e., the statement that ).

Thus, if we assume that there is a non-even function of order 1, then we can conclude that . Can we weaken the hypothesis and still conclude something interesting? We can, as they also proved:

For any , if there is a non-even function of order , then .