Let us define the *Parovichenko cardinal* $ \mathfrak{P}$ as the smallest cardinal $ \kappa$ such that each compact Hausdorff space $ K$ of weight $ w(K)<\kappa$ is the continuous image of the remainder $ \beta\mathbb N\setminus\mathbb N$ of the Stone-Cech compactification of the discrete space of positive integers $ \mathbb N$ .

By a classical theorem of Parovichenko, $ \mathfrak P\ge\aleph_2$ .

On the other hand, Theorem 2.7 of this paper of van Douwen and Przymusinski implies that $ \mathfrak P\ge\mathfrak p$ where $ \mathfrak p$ is the well-known pseudointersection number.

These two results imply that $ \mathfrak P\ge\max\{\aleph_2,\mathfrak p\}$ .

So, under CH we have $ \mathfrak P=\aleph_2>\mathfrak c=\mathfrak p=\aleph_1$ .

Alan Dow and KP Hart observed (on the page 1833 of their paper in TAMS) that in the Cohen model $ \mathfrak P=\aleph_2=\mathfrak c>\mathfrak p=\aleph_1$ .

Finally, PFA implies $ \mathfrak P=\aleph_2=\mathfrak c=\mathfrak p>\aleph_1$ , see Corollary 4.6 in Baumgartner’s survey “Applications of the Proper Forcing Axiom” in the “Handbook of Set-Theoretic Topology”.

Problem.Is it consistent that $ \mathfrak P>\max\{\aleph_2,\mathfrak p\}$ ?