I read [here](https://www.researchgate.net/post/The_most_important_open_problems_in_set_theory2) that the following problem of Hausdorff is apparently still open.



> 
> Is there a maximal branch $C$ in the poset $\omega^\omega$ with the eventual domination order, such that $C$ has no $\omega\_1$-gap.
> 
> 
> 


My questions concerning the above problem are:


1. What is its current status?
2. What are the consequences of its solution? negative, independence or positive.



Hausdorff's question on $\omega_1$-gap
I think the problem is solved. See


[Francisco Kibedi: Maximal Saturated Linear Orders](http://settheory.mathtalks.org/francisco-kibedi-maximal-saturated-linear-orders/)


See also Kibedi's PhD thesis 


[Maximal Saturated Linear Orders](https://yorkspace.library.yorku.ca/xmlui/handle/10315/32107).




---


The problem is independence of ZFC. The question asks "is there a pantachy containing no $(\omega\_1, \omega\_1^\*)$-gap?". See Kanovei's answer given in [Hahn's Embedding Theorem and the oldest open question in set theory](https://mathoverflow.net/questions/128935/hahns-embedding-theorem-and-the-oldest-open-question-in-set-theory).


Now as the author explains, CH implies a no answer to the above equivalent question (known to Hausdorff). The author shows the consistency of the statement. 


