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> Let us define the *Parovichenko cardinal* $\mathfrak{P}$ as the largest 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 $\mathbb N$. 
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By a classical theorem of Parovichenko, $\mathfrak P\ge\aleph\_2$. 


On the other hand, Theorem 2.7 in [this paper](http://matwbn.icm.edu.pl/ksiazki/fm/fm105/fm105112.pdf) of van Douwen and Przymusinski implies that $\mathfrak P\ge\mathfrak p$ where $\mathfrak p$ is the well-known [pseudointersection number](http://www.math.lsa.umich.edu/~ablass/hbk.pdf). 



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> These two results yield the inequality $\mathfrak P\ge\max\{\aleph\_2,\mathfrak p\}$. 
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So, under CH we have $\mathfrak P=\aleph\_2>\mathfrak c=\mathfrak p=\aleph\_1$.


By a result of Kunen of 1968 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](https://www.sciencedirect.com/science/article/pii/B9780444865809500240)" in the "Handbook of Set-Theoretic Topology".


Let us also mention a result that follows from Theorems 2.1 and 2.2 of [van Douwen and Przymisinski](http://matwbn.icm.edu.pl/ksiazki/fm/fm105/fm105112.pdf):


$\mathfrak P\le\mathfrak c$ if one of the following conditions holds:


$\bullet$ $\aleph\_2\le\mathfrak c<2^{\aleph\_1}=\aleph\_{\omega\_2}$ or


$\bullet$ $\mathfrak c=2^{\aleph\_1}$ and $\mathfrak q\_0=\aleph\_1$.


Here [$\mathfrak q\_0$](http://journals.iapmm.lviv.ua/ojs/index.php/MBSSS/article/view/522/974) is the largest cardinal $\kappa$ such that each subset $X\subset \mathbb R$ of cardinality $|X|<\kappa$ is a $Q$-set (which means that each subset of $X$ is an $F\_\sigma$-set in $X$).



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> **Problem.** Is it consistent that $\mathfrak P>\max\{\aleph\_2,\mathfrak p\}$?
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The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?
No -- it is consistent that $\mathsf{CH}$ fails, and that every compact Hausdorff space of weight $\leq\!\mathfrak{c}$ is a continuous image of $\beta \mathbb N \setminus \mathbb N$. (This is due to Baumgartner, who mentions it off-hand in his article in the Handbook of Set Theoretic Topology; the mutual consistency with $\mathsf{MA}\_{\sigma\text{-linked}}$ was established later by Baumgartner, Frankiewicz, and Zbierski in [this paper](http://matwbn.icm.edu.pl/ksiazki/fm/fm136/fm136119.pdf).) In such a model, $\mathfrak{P} = \mathfrak{c}^+ > \max\{\aleph\_2,\mathfrak{p}\}$.


