The Type $\omega$-Vaught's Conjecture for $\omega$-stable Theories

We propose a new variant of Vaught’s conjecture, based on Martin’s (model-theoretic) conjecture and the $\omega$-Vaught’s conjecture of Gonzalez and Montalbán. For $\omega$-stable theories, we prove the conjecture by analyzing in detail Bouscaren’s and Shelah-Harrington-Makkai’s proofs of Martin’s conjecture and Vaught’s conjecture, respectively, for $\omega$-stable theories. In particular, when there are countably many countable models, we improve the required number of quantifiers for a Scott sentence from $\omega +\omega$ to $\omega +5$, and we obtain a similar bound of $\omega +6$ for the uncountable case.