Isomorphism of computable structures and Vaught's Conjecture

Abstract

The following question is open: Does there exist a hyperarithmetic class of computable structures with exactly one non-hyperarithmetic isomorphism-type? Given any oracle $a \in 2^\omega$, we can ask the same question relativized to $a$. A negative answer for every $a$ implies Vaught's Conjecture for $L_{\omega_1 \omega}$.