On minimal counterexamples to Vaught’s Conjecture

Let [Formula: see text] be an [Formula: see text] sentence which is a minimal counterexample to Vaught’s Conjecture and for [Formula: see text] let [Formula: see text] be the fragment of [Formula: see text] consisting of formulas with quantifier rank less than [Formula: see text]. For a cub set of [Formula: see text], for all countable models [Formula: see text] and [Formula: see text] of [Formula: see text] with Scott rank at least [Formula: see text], for all [Formula: see text], [Formula: see text] is [Formula: see text]-embeddable into [Formula: see text].