175 Notre Dame Journal of Formal Logic Volume 33,  Number 2,  Spring 1992 On Generic Structures D.W. KUEKER and M. C.  LASKOWSKI Abstract We discuss many generalizations of Fraisse's construction  of countable 'homogeneous universal' structures. We give characterizations of when such a structure is saturated and when  its theory is ω categorical. We also state very general conditions under which the structure is atomic. / zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Introduction In  this paper we investigate variations on the classical con struction  of countable homogeneous universal structures from appropriate classes of finite structures. The most basic result here is the following theorem of Fraisse [1]: Theorem 1.1zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Let K be a class of finite structures in a finite, relational lan guage that is closed under isomorphism and substructure. Assume further that K satisfies the joint embeddingproperty and amalgamation. Then, 1. there is a unique, countable Q which is "homogeneous universal" for K, i.e., a is (ultra) homogeneous and K is precisely the class of finite structures em beddable in d; 2. the complete theory of the structure (I in (1) is ω categorical. It  is easy to see that (1) holds also for countably infinite relational languages provided K contains only countably many isomorphism types, but (2) may fail in  this context. If K is not closed under substructure then the same basic argu ment  establishes a variant of (1) in which  CE satisfies a weaker sort of homoge neity (called pseudo homogeneity by Fraisse); here too (2) may fail, even if the language is finite. More recently, Hrushovski [3,4] has used a construction that generalizes the basic construction by replacing substructure by stronger relations. In  this paper we unify all of these variations in a single framework (allow ing also functions and constants in the language). We refer to the resulting struc tures as generic rather than as homogeneous universal. We then investigate some properties of these generics. Ever since Morley Vaught there has been a tendency to view homogeneous universal structures as analogues of saturated models. The Received December 3, 1990; revised January 20, 1992