Given a cardinal k, what's the biggest dense linear order with a dense subset of size k?

Question

It's not hard to show that for any cardinal $\kappa$, there is no dense linear order without endpoints (DLO) of size greater than $2^{\kappa}$ that has a dense subset of size $\kappa$. But one can show that if $\mu$ is the least cardinal such that $\kappa^{\mu} > \kappa$, then there's a DLO of size $\kappa^{\mu}$ with a dense subset of size $\kappa$, so in particular there's a dense linear order of size greater than $\kappa$ with a dense subset of size $\kappa$. Thus in models of GCH, the answer to the question in the title is "as big as possible." Also from ZFC alone, the case $\kappa = \omega$ also has the answer "as big as possible": just consider $\mathbb{Q}$ and $\mathbb{R}$.

So the question is: Is it consistent that for some $\kappa$, there is no DLO of size $2^{\kappa}$ with a dense subset of size $\kappa$?

Apparently the answer's supposed to be "yes," and the relevant notion is that of a "Dedekind number," but searches for "Dedekind number" haven't yielded anything relevant, so if someone could point me to a relevent reference, that'd be great too.

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Unless I made a mistake, I've shown that a DLO $(L,\leq)$ is a DLO with a dense subset of size $\kappa$ iff it is (up to isomorphism) some set $X \subset \mathcal{P}(\kappa)$ ordered by inclusion such that $(X,\subseteq)$ satisfies the following property:

$(*)\ \ \forall \alpha < \kappa$, the set $\{ x \in X : \alpha \notin x \}$ has a least upper bound, let's call it $x _{\alpha}$, in $(X,\subseteq)$

Furthermore, I've shown that given $X \subset \mathcal{P}(\kappa)$ which is a DLO under inclusion (and $\cup X = \kappa$, and $|X| > \kappa$), it can be modified to give $X' \subset \mathcal{P}(\kappa)$ which is also a DLO under inclusion, has the same size as $X$, and has a dense subset of size $\kappa$: First, obtain $X_0$ from $X$ by removing, for each $\alpha < \kappa$, the greatest lower bound in X (if it exists) of the collection $\{x \in X : \alpha \in x\}$; Next obtain $X'$ from $X_0$ by adding, for each $\alpha < \kappa$, the set $x _{\alpha} = \cup \{x \in X_0 : \alpha \notin x\}$.

So an equivalent reformulation of the question is: Is it consistent that for some $\kappa$, there is no $X \subset \mathcal{P}(\kappa)$ of size $2^{\kappa}$ which forms a DLO under inclusion?

Answer 1

The supremum of the cardinalities of linear orders with a dense subset of size $\kappa$ is called $\rm{ded}(\kappa$) in Jerry Keisler's paper "Six classes of theories" (J. Australian Math. Soc. 21 (1976) 256-266), where it (along with its $\omega$th power) is part of the answer to a fundamental question in stability theory. I don't have access to the paper at the moment, but the review on MathSciNet mentions that (Keisler mentions that) Bill Mitchell showed the consistency of $(\rm{ded}(\omega_1))^\omega<2^{\omega_1}$; the reference given there is Annals of Math. Logic 5 (1972) 21-46.

Answer 2

This is very late, but in 2012, Artem Chernikov, Itay Kaplan and Saharon Shelah posted on arXiv a paper titled "On non-forking spectra" ( http://arxiv.org/abs/1205.3101 ). They claim that it is consistent that $Ded(\kappa)< Ded(\kappa)^\omega$. In particular, it is consistent that $Ded(\kappa)<2^\kappa$. It is still open whether both inequalities can be made strict: $Ded(\kappa)<Ded(\kappa)^\omega<2^\kappa$.

In their paper they cite both Keisler's and Mitchell's paper that Andreas Blass metnions in his answer. So, as far as I know this is the most recent development that relates to your question.