DOP and FCP in generic structures

John T. Baldwin, Saharon Shelah
1998 Journal of Symbolic Logic (JSL)  
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 −α. In [2] we proved that this was the same theory as the
more » ... me theory as the theory T α 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 just the T α. The basic facts cited from [3] were due to Hrushovski [4]; a full bibliography is in [3]. For general background in stability theory see [1] or [5]. 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.
doi:10.2307/2586841 fatcat:6rahqu2uijfsrj6z4v4mgwt4uq