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.

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?**

`yes,' and the relevant notion is that of a`

Dedekind number,'" ? $\endgroup$ – Ed Dean Dec 4 '10 at 3:26