# Hausdorff's question on $\omega_1$-gap

I read here 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.

I think the problem is solved. See

Francisco Kibedi: Maximal Saturated Linear Orders

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.
• The abstract is not exhaustively clear. The key result is formulated as Con(ZFC + ¬CH + ∃ a maximal saturated linear order of size continuum in the space of real-valued sequences partially ordered by eventual domination), where saturated' lacks the cardinal parameter of saturation, while of size continuum' looks strange since the whole poset is of cardinality c. Also, there is no indication how the key result implies a required gapless pantachy. – Vladimir Kanovei Apr 4 '17 at 19:34