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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?

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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.

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    Are you referring to the ​ "We use it in a way similar to Martin-Solovay [12] who showed that assuming ..." ​ sentence? ​ ​ ​ If no, then where in that paper is your answer's starting sentence explained? ​ ​ ​ ​ ​ ​ ​ ​ – 2017-02-07
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    It is shortly after that, just after the proof of Lemma 3.8 (keywords: self-constructible reals). – 2017-02-07
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    Do you know of any reference for their "equiconsistent" statement? ​ (top of page 238) ​ ​ ​ ​ – 2017-02-07
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    @Ricky: Just do the usual forcing to get Martin's Axiom over $L$. That forcing is ccc, so it preserves cardinals. – 2017-02-08