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Definition If there is a dense linear order w/o endpoints of size $\lambda$ with a dense subset of size $\kappa$ then write $D(\kappa,\lambda)$. $Ded(\kappa)=\sup_\lambda \{D(\kappa,\lambda)\}$.

It is known that both $Ded(\kappa)=2^\kappa$ and $Ded(\kappa)<2^\kappa$ are consistent.

Do you have a reference where (consistently) $Ded(\kappa)$ is not attained, i.e. it is a supremum rather than a maximum?

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  • $\begingroup$ Reviewing the literature and asking people around, my conclusion is that the problem has not been studied (2014/10). Of course, there is always the possibility of a paper that answers this questions and I did not work hard enough to find. $\endgroup$
    – Ioannis Souldatos
    Oct 30, 2014 at 14:03

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