A non-principal ultrafilter $\mathcal{U}$ on $\omega$ is called *selective* (or *Ramsey*) if for every partition $\mathcal{P}$ of $\omega$ disjoint with $\mathcal{U}$ there is $A\in\mathcal{U}$ such that $|A\cap B|\le1$ for every $B\in\mathcal{P}$. There are several weakenings of this definition, but still expressing some kind of selectivity; most notably *P-points* and *Q-points*.

My question is as follows. Do you know any notion of *highly non-selective* ultrafilters? A class of ultrafilters about which we can say that they are *very far* from being selective? Or maybe some way of measuring *how* ultrafilters are selective?

Selective ultrafilters are exactly the minimal points of the Rudin-Keisler ordering of non-principal ultrafilters on $\omega$, so any ultrafilter which is not a minimal one is not selective. But how can we see this ordering in the context of selectivity?

I know that the question is vague, but maybe you have some association with it. I will appreciate any answer.