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 .