Given two set $A,B$ we write $A\subset^\* B$ if the complement $A\setminus B$ is infinite.




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A *Hausdorff gap* is a transfinite family $\langle A\_\alpha,B\_\alpha\rangle\_{\alpha\in\omega\_1}$ of infinite subsets of $\omega$ satisfying the following two properties:


$\bullet$ $A\_\alpha\subset^\* A\_\beta\subset^\* B\_\beta\subset^\* B\_\alpha$ for any countable ordinals $\alpha\le\beta$;


$\bullet$ for any infinite subset $I\subset \omega$ there exists a countable ordinal $\alpha$ such that $A\_\alpha\not\subset^\* I$ or $I\not\subset^\* B\_\alpha$.




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A in infinite subset $I\subset\omega$ is called a *pseudointersection* of a Hausdorff gap $\langle A\_\alpha,B\_\alpha\rangle\_{\alpha\in\omega\_1}$ if $I\subset^\* (B\_\alpha\setminus A\_\alpha)$ for any $\alpha\in\omega\_1$.




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[The definition](https://en.wikipedia.org/wiki/Pseudo-intersection) of the small uncountable cardinal $\mathfrak p$ implies that under $\mathfrak p>\omega\_1$ each Hausdorff gap has an infinite pseudointersection. 


What does happen under $\mathfrak p=\omega\_1$?



> 
> **Problem.** Is $\mathfrak p=\omega\_1$ equivalent to the existence of a Hausdorff gap without infinite pseudointersection?
> 
> 
> 



Is $\mathfrak p=\omega_1$ equivalent to the existence of a Hausdorff gap without infinite pseudointersection?
Yes. This is a result due to Nyikos and Vaughan from 1983, appearing the paper


*Nyikos, Peter J.; Vaughan, Jerry E.*, [**On first countable, countably compact spaces. I: ((\omega\_ 1,\omega^\*\_ 1))-gaps**](http://dx.doi.org/10.2307/1999547), Trans. Am. Math. Soc. 279, 463-469 (1983). [ZBL0542.54004](https://zbmath.org/?q=an:0542.54004).


Hausdorff gaps without a pseudo-intersection are called *tight* gaps, and Theorem 1.2 of the paper shows that these exist if and only if $\mathfrak{p}=\omega\_1$.


