# Models of $\mathsf{ZFC}$ with neither $P$- nor $Q$-points

A $$P$$-point is an ultrafilter $$\scr U$$on $$\omega$$ such that for every function $$f:\omega\to\omega$$ there is $$x\in {\scr U}$$ such that the restriction $$f|_x$$ is either constant, or finite-to-one.

A $$Q$$-point is an ultrafilter $$\scr U$$on $$\omega$$ such that for every function $$f:\omega\to\omega$$ with the property that $$f^{-1}(\{m\})$$ is finite for each $$m\in \omega$$, there is $$x\in {\scr U}$$ such that the restriction $$f|_x$$ is injective.

$$P$$-points need not exist, and $$Q$$-points need not exist.

Question. Is it possible that neither $$P$$- nor $$Q$$-points exist?

As far as I know, the consistency of the statement "neither P-points nor Q-points exist" is still an open problem. It is known that this statement implies $$2^{\aleph_0}\geq\aleph_3$$.
The results involve the cardinal characteristics cov$$(\mathcal B)$$ [the smallest number of meager sets to cover $$\mathbb R$$], $$\mathfrak d$$ [the smallest number of functions $$\omega\to\omega$$ to dominate all such functions], and $$\mathfrak c$$ [the cardinal of the continuum]. It is provable in ZFC that $$\aleph_1\leq\text{cov}(\mathcal B)\leq\mathfrak d\leq\mathfrak c.$$ Theorem 9.25 says (among other things) that if $$\mathfrak d=\mathfrak c$$ then there exists a P-point. Theorem 9.27 says (among other things) that if cov$$(\mathcal B)=\mathfrak d$$ then there exists a Q-point. So in order to have neither a P-point nor a Q-point, you'd need $$\aleph_1\leq\text{cov}(\mathcal B)<\mathfrak d<\mathfrak c,$$ which implies $$\mathfrak c\geq\aleph_3$$.