How "simple" can a counterexample to the Continuum Hypothesis be?

Question

For this question, work in your choice of ZFC or ZF+DC. ​ ​ ​ ​ ( CH ​ = ​ the Continuum Hypothesis )

Analytic sets are known to have the perfect set property, so they cannot be counterexamples to CH. ​ Is there any consistency result regarding co-analytic counterexamples to CH?
If yes, what about with bounds on the complexity of the
continuous function from the Baire space onto the set's complement?
For example:

Can the function's graph be hyperarithmetical?
Can the function be computable in the sense of receiving the input as an oracle?

Answer 1

It is consistent that there is a coanalytic set (i.e., $\Pi^1_1$) which has size $\aleph_1$, while the continuum is arbitrarily large.

You can find this explained in:

Greg Hjorth, Leigh Humphries, and Arnold W. Miller, MR 3087073 Universal sets for pointsets properly on the $n$th level of the projective hierarchy, J. Symbolic Logic 78 (2013), no. 1, 237--244.