The spectrum problem I: אɛ-satuarted models, the main gap

The main goal of this paper is to complete the classification of those first-order theories such that {Mathematical expression}. We introduce two notions, the dimensional order property and deepness. Our main theorem asserts that for a superstable theory {Mathematical expression} iff T has the dimensional order property or is deep. In a sense made precise in §4 this provides a syntactical characterization of theories with the maximum number of אe{open}-saturated models in each power.