Is it true (in ZFC) that for any regular infinite cardinal $\kappa$ there exists an ordered field of weight $\kappa$ and cardinality $2^\kappa$ (or at least $>\kappa$)?
The field of real numbers shows that for $\kappa=\aleph_0$ the answer is trivially "yes". It seems that this question has an affirmative answer in the Constructive Universe. Is it true in ZFC?
Or the converse is true: there is a model of ZFC in which all ordered fields of weight $\aleph_1$ have cardinality $\aleph_1$?
Browsing through MathOverflow I have found some answers to related questions, which shed light on my question too. It turned that my question has been considered in the literature. The most appropriate sources are: H.J. Keisler, Six classes of theories, J.Austral Math. Soc. 21 (1976), 257-266, A. Chernikov, I. Kaplan, S. Shelah: http://arxiv.org/abs/1205.3101, and A. Chernikov, S. Shelah: http://arxiv.org/abs/1308.3099. According to Keisler, the supremum of cardinalities of ordered fields containing a dense subset of size $\kappa$ is equal to $ded(\kappa)$ the supremum of cardinalities of linearly ordered spaces containing a dense subset of size $\le\kappa$. The cardinal $ded(\kappa)$ is contained in the interval $(\kappa,2^\kappa]$ and it is consistent that $ded(\kappa)<2^\kappa$ for any cardinal $\kappa$ of uncountable cofinality. On the other hand, by a recent result of Chernikov and Shelah (cited above), $2^\kappa\le ded(ded(ded(ded(\kappa))))$ (which is more than sufficient for my purposes).
On the one hand, if $K$ is an ordered field of weight $\kappa$, then trivially
$$|K|\le\mathrm{ded}(\kappa):=\sup\bigl\{|L|:\text{$L$ is a linear order with dense subset of size $\kappa$}\bigr\}\le2^\kappa.$$
On the other hand, we have the following lower bounds, which in particular show that there is always an OF of weight $\kappa$ and cardinality strictly larger than $\kappa$.
Proposition 1: If $\lambda$ is the least cardinal such that $\kappa^\lambda>\kappa$, there exists an ordered field of cardinality $\kappa^\lambda$ with a dense subfield of cardinality $\kappa$.
Proof: Note that the condition implies that $\lambda$ is regular, and $\lambda\le\kappa$.
First, we construct an increasing sequence of fields $\{F_\alpha:\alpha\le\lambda\}$ of cardinality $\kappa$ as follows. Let $F_0$ be an OF of cardinality $\kappa$ such that $(0,1)$ contains $\kappa$ disjoint nonempty intervals. (Incidentally, it’s easy to show that an OF of weight $\kappa$ always contains $\kappa$ disjoint intervals, but we will not need this.) If $\alpha$ is limit, we put
$$F_\alpha=\bigcup_{\beta<\alpha}F_\beta.$$
For the successor step, since $\kappa^{<\lambda}=\kappa$, there are only $\kappa$ pairs of subsets $A,B\subseteq F_\alpha$ such that $A<B$, and $|A|,|B|<\lambda$. Thus, using the compactness theorem, there exists an extension $F_{\alpha+1}\supseteq F_\alpha$ of size $\kappa$ such that for every such $A,B$, there is an element $c\in F_{\alpha+1}$ with $A<c<B$, and moreover, there is an element $u\in F_{\alpha+1}$ such that $u>F_{\alpha}$.
Using the regularity of $\lambda$, the field $F:=F_\lambda$ thus constructed satisfies:
$|F|=\kappa$,
$(0,1)_F$ contains $\kappa$ disjoint (nondegenerate) intervals,
$F$ has an increasing cofinal subsequence $\{u_\alpha:\alpha<\lambda\}$,
$F$ has the $\eta_\alpha$ property for $\aleph_\alpha=\lambda$, that is, if $A,B\subseteq F$ are such that $A<B$ and $|A|,|B|<\lambda$, there is $c\in F$ such that $A<c<B$.
Let $\hat F$ be the Scott completion of $F$, which is the largest ordered field extension of $F$ in which $F$ is dense. It suffices to show that $|\hat F|\ge\kappa^\lambda$, i.e., in the terminology of https://mathoverflow.net/a/140962, that $F$ has at least $\kappa^\lambda$ good cuts.
We can construct a tree $\{I_t:t\in\kappa^{<\lambda}\}$ of nondegenerate intervals $I_t=[a_t,b_t]$ so that:
If $s$ properly extends $t$, then $a_t<a_s<b_s<b_t$.
If $t$ and $s$ are incomparable, $I_t\cap I_s=\varnothing$.
If $\mathrm{dom}(t)=\alpha<\lambda$, then $b_t-a_t<1/u_\alpha$.
We build the tree by induction on $\mathrm{dom}(t)$. For $I_\varnothing$, we can take any interval shorter than $1/u_0$. For the successor step, if $I_t$ has already been constructed, where $\mathrm{dom}(t)=\alpha$, we take the sequence of intervals from (2), scale it down into a subinterval of $I_t$ shorter than $1/u_{\alpha+1}$, and call it $\{I_{t_\smile\beta}:\beta<\kappa\}$. Finally, if $\alpha$ is limit, then $$A:=\{a_{t\restriction\beta}:\beta<\alpha\}< B:=\{b_{t\restriction\beta}:\beta<\alpha\}$$ have size $<\lambda$, hence applying (4) twice, we can find $I_t=[a_t,b_t]$ so that $A<a_t<b_t<B$.
Now, the properties of the tree ensure that for any $\tau\in\kappa^\lambda$, the sets $$A_\tau=\bigcup_{\alpha<\lambda}(-\infty,a_{\tau\restriction\alpha}],\qquad B_\tau=\bigcup_{\alpha<\lambda}[b_{\tau\restriction\alpha},+\infty)$$ form a good cut, and $(A_\tau,B_\tau)\ne(A_{\tau'},B_{\tau'})$ for $\tau\ne\tau'$.
Note that if we slightly modify the construction of $F_\alpha$ so that we also take a real closure on each step, then $F$ becomes real closed, in which case $\hat F$ is also real closed.
Corollary 2: If $\nu$ is the least cardinal such that $2^\nu>\kappa$, there exists an ordered (real-closed) field of cardinality $2^\nu$ with a dense subfield of cardinality $\kappa$.
Proof: Put $\mu=2^{<\nu}\le\kappa$ and $\lambda=\mathrm{cf}(\nu)$. An exercise in cardinal arithmetic shows that $\mu^{<\lambda}=\mu$ and $\mu^\lambda=2^\nu$, hence there exists a RCF of size $2^\nu$ with a dense subset of size $\mu$, which we can enlarge to $\kappa$.
Note that for a given $\kappa$, the bound in the Corollary is better than in the Proposition: clearly $\lambda\le\nu$, hence $\kappa^\lambda\le(2^\nu)^\lambda=2^\nu$.
