We investigate the class of m -hypergraphs in which substructures with l elements have more than s subsets of size m that do not form a hyperedge. The class has a (unique) Fraïssé limit, if 0 ≤ s < ( l − 2 m − 2 ) . We show that the theory of the Fraïssé limit has SU -rank one if 0 ≤ s < ( l − 3 m − 3 ) , and dividing and forking will be different concepts in the theory if ( l − 3 m − 3 ) ≤ s < ( l − 2 m − 2 ) .

Abstract Let be a finite relational language andα= (αR:R∈ ) a tuple with 0 <αR≤ 1 for eachR∈ . Consider a dimension function where eacheR(A)is the number of realizations ofRinA. LetKαbe the class of finite structuresAsuch thatδα(X)≥ 0 for any substructureXofA. We show that the theory of the generic model ofKαis AE-axiomatizable for anyα.

Abstract If T is a model complete theory with the strict order property, then the theory of the models of T with an automorphism has no model companion.

Abstract For a theory T in L, Tσ is the theory of the models of T with an automorphism σ. If T is an unstable model complete theory without the independence property, then Tσ has no model companion. If T is an unstable model complete theory and Tσ has the amalgamation property, then Tσ has no model companion. If T is model complete and has the fcp, then Tσ has no model completion.

Let us consider countable languages L containing a unary predicate symbol P and L− =L\{P}. We also assume that L is relational. Then for any L-structure M, N = PM can naturally be considered as an L−-substructure of M. The main object of this paper will be the study of the following question: Under what condition does M have to be ℵ0-categorical. ℵ1-categorical, or stable if N is? Hodges and Pillay [6] proved that if M is a countable symmetric extension of N and T = Th(M) is minimal over P (they said that T is one-cardinal over P), then the total categoricity of N implies that of M. This is a solution to a problem in Ahlbrandt and Ziegler [1]. The condition that “M is a symmetric extension of N” is an interpretation of the condition “every relation on N definable in M is definable within N”. We shall give several interpretations of this phrase: They are the Ø-reduction property, the reduction property, the strong reduction property, and the uniform reduction property (Definition 1). Under the assumptions, we study the question proposed above. In §3 we treat the case that M is countable and show that if T is minimal over P and M has the strong reduction property over N, then M is ℵ0-categorical if N is (Theorem 5). This is a slight extension of the result of Hodges and Pillay mentioned above. (If M is countable and saturated, then the strong reduction property is equivalent to the condition that M will be symmetric over N if we add a finite number of appropriate constants.) A counterexample to this theorem has been obtained by Hrushovski in the case that only the Ø-reduction property is assumed. We also give a stronger result: If M has the Ø-reduction property over N and is ℵ0-categorical, M\N is infinite, and N is algebraically closed, then there is an expansion M* of M such that M* is not ℵo-categorical but M* still has the Ø-reduction property over N (Theorem 6). Moreover, we give an example such that M has the uniform reduction property over N. Th(M*) is minimal over P. N is ℵ0-categorical but M is not.