DOP and FCP in Generic Structures

We work throughout in a finite relational language L. This paper is built on [2] and [3]. We repeat some of the basic notions and results from these papers for the convenience of the reader but familiarity with the setup in the first few sections of [3] is needed to read this paper. Spencer and Shelah [6] constructed for each irrational α between 0 and 1 the theory T α as the almost sure theory of random graphs with edge probability n. In [2] we proved that this was the same theory as the theory Tα built by constructing a generic model in [3]. In this paper we explore some of the more subtle model theoretic properties of this theory. We show that T α has the dimensional order property and does not have the finite cover property. We work in the framework of [3] so probability theory is not needed in this paper. This choice allows us to consider a wider class of theories than Partially supported by NSF grant 9308768. This is paper 567 in Shelah’s bibliography. Both authors thank Rutgers University and the Binational Science Foundation for partial support of this research.