We prove that, for every theory $T$ which is given by an ${mathcal L}_{omega_1,omega}$ sentence, $T$ has less than $2^{aleph_0}$ many countable models if and only if we have that, for every $Xin 2^omega$ on a cone of Turing degrees, every $X$-hyperarithmetic model of $T$ has an $X$-computable copy. We also find a concrete description, relative to some oracle, of the Turing-degree spectra of all the models of a counterexample to Vaughts conjecture.
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