DOP and FCP in generic structures
Journal of Symbolic Logic (JSL)
We work throughout in a finite relational language L. This paper is built on  and . 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  is needed to read this paper. Spencer and Shelah  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 −α. In  we proved that this was the same theory as the
... me theory as the theory T α built by constructing a generic model in . 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  so probability theory is not needed in this paper. This choice allows us to consider a wider class of theories than just the T α. The basic facts cited from  were due to Hrushovski ; a full bibliography is in . For general background in stability theory see  or . We work at three levels of generality. The first is given by an axiomatic framework in Context 1.10. Section 2 is carried out in this generality. The main family of examples for this context is described in Example 1.3. Sections 3 and 4 depend on a function δ assigning a real number to each finite L-structure as in these examples. Some of the constructions in Section 3 (labeled at the time) use heavily the restriction of the class of examples to graphs. The first author acknowledges useful discussions on this paper with Sergei Starchenko.