## The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?

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 $$\mathfrak P>\max\{\aleph_2,\mathfrak p\}$$?