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.

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