Cardinality of definable sets of reals Throughout this question we assume ZFC.
If CH holds, then the following is obvious:

(S) Every definable infinite subset of $\mathbb R$ has size either $\aleph_0$ or $2^{\aleph_0}$.

(It's true because every subset of reals satisfies this, in particular so does every definable)
Recently my friend has asked me whether the same is true if we assume CH fails. My answer was that this is independent of ZFC, because I'm fairly sure that results about pointwise definable models will gives us such model of ZFC + not CH (correct me if I'm wrong!) in which a set of reals of size $\aleph_1$ is definable, and I can recall seeing a result that (S) can hold, possibly under some large cardinal assumption. However, I failed to find a reference for that.
Hence my question is for a reference of the following result (if my memory isn't failing me and it's actually true):

It is consistent (relatively to large cardinals) that CH fails and (S) holds.

I also wanted to ask if this is true if CH fails badly:

Is it consistent for every cardinal $\kappa$ that (S) holds and $2^{\aleph_0}>\kappa$?

Thanks in advance.
 A: EDIT: This isn't really an answer - see Andres' comment - but you might find it interesting, and it is too long for a comment.
Your second question is vague, but has an affirmative answer in the following sense:

Let $V\models ZFC+V=L$, and let $\kappa$ be a cardinal in $V$. Then there is a forcing extension $V[G]$ of $V$ in which $2^{\aleph_0}>\kappa$ but there is a definable set of reals of cardinality $\aleph_1$.

Proof: Add $\kappa^+$-many Cohen reals; this bumps the continuum up to $\kappa^+$. Meanwhile, it does not collapse $\aleph_1$, so the set $V\cap\mathbb{R}$ of ground reals is still of size $\aleph_1$. And $V\cap\mathbb{R}=(\mathbb{R}^L)^V=(\mathbb{R}^L)^{V[G]}$, so is definable in $V[G]$ without parameters.

The first question seems harder. Any countable model of ZFC has a class forcing extension which is pointwise definable, but this class forcing extension doesn't seem to preserve large cardinal properties. Meanwhile, "big enough" large cardinals kill off all the obvious ways to try to define sets of reals of intermediate cardinality.
