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 The notion of quasi-minimal structures was denned by B. Zil'ber as a natural generalization of minimal structures. Inspired by his work, we study here basic model theoretic properties of quasi-minimal structures. Main result is the construction of ω-saturated quasi-minimal models under ω-stability assumption.

Let L0, L1 and L2 be countable languages with L ∩ L1 = L0. Let M0 be an L0-structure and Mi, an expansion of M0 to an Li,-structure (i = 1,2). We will call an L1 ∪ L2-structure M an amalgamation of M1 and M2 if M∣Li ≅ Mi, (i = 1,2). Let's consider the following problem. (*) Suppose that both M1 and M2 belong to the class . Can we always find an amalgamation M in ? Of course the existence of such an amalgamation depends on the class L. Some examples of and the answers are given below. 1. = Countably saturated strongly minimal structures with the DMP In [3], Hrushovski showed that any two strongly minimal theories formulated in totally different languages have a common extension which is still strongly minimal and with the DMP (DMP is the property that states that if a point is sufficiently close to ā, then φ( , ) has the same rank and the same degree as φ( , ā).) His proof essentially shows that if L0 = ∅ then any two countably saturated strongly minimal structures with the DMP have a strongly minimal amalgamation. Also he gave an example that shows the condition L0 = ∅ is necessary. 2. = ℵ1-categorical countable structures. Let M1 be the structure (ℚ, +) and let M2 be the {E, F}-structure defined by: (i) E is an equivalence relation which divides the universe into two infinite classes A and B, (ii) F is a bijection between A and B.

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.

Galois theory states that if L is a certain algebraic extension (called a Galois extension) of a field K, then there is a one-to-one correspondence (called a Galois correspondence) between subfields M, K ⊂ M ⊂ L and subgroups of the automorphism groups of L fixing the elements in K. A subfield of a field L can be considered as a substructure of L in general model theory. However, a substructure is a subset closed under functions which are interpretations of function symbols in a given language, so the notion of substructure may change if we expand the language by adding definable notions. On the other hand a definably closed substructure is a subset which is closed under all definable functions, and it does not change by such expansions. If we are interested in subfields of an algebraically closed field of characteristic 0, these two notions are the same. But in a field of prime characteristic they are not equal. Speaking roughly, a Galois extension of a field K is an extension whose subfields are relatively definably closed. Poizat [4] showed that if a structure M has elimination of imaginaries there is a kind of Galois correspondence between definably closed substructures and subgroups of bijective elementary mappings of M. In this paper, using Poizat's result we study algebraic types. As is well known, one motivation for developing the Galois theory was to show the unsolvability of equations with degree ≥ 5. We want to take this unsolvability as a special case of general phenomena. For this purpose, we introduce several notions which are stronger than mere algebraicity and study relations between these notions and groups of bijective elementary mappings. (See Theorems 3.7 and 3.9.)

In [8], we have shown the equivalence of almost strong minimality and strong unidimensionality. More precisely, we proved: Theorem [8]. Let T be a countable stable theory. Then the following two conditions are equivalent: (i) T is almost strongly minimal; (ii) T can be extended to a theory such that any two nonalgebraic types are not almost orthogonal. In the present paper, we define the notion of strong 2-dimensionality (of T). We show that if T is strongly 2-dimensional then T is ω-stable and its model has a simple structure. Roughly speaking, in a model of a strongly 2-dimensional theory, one of the following holds: (a) every element is in acl (δi is strongly minimal), or (b) every element is in acl (δ is strongly regular). Shelah's definition of 2-dimensionality does not imply even superstability. (See Exercise 5.5 in [6, Chapter V, §5].) We show also that condition (a) above implies strong 2-dimensionality of T. However condition (b) does not imply strong 2-dimensionality in general. Our notations and conventions are standard. T is always countable and stable. We work in . A,B,… are used to denote small subsets of . , … are used to denote finite sequences of elements in . δ, φ,… are used to denote formulas (with parameters), p, q, … are used to denote types (with parameters). The fact that p is a nonforking (forking) extension of q is denoted by p ⊃nfq(p ⊃fq). If p is stationary, p∣A denotes the type in S(A) which is parallel to p. (or ) denotes the set of realizations of p (or δ). The Morley rank of p is denoted by RM(p).

In this paper we shall state some interesting facts concerning non-ω-categorical theories which have only finitely many countable models. Although many examples of such theories are known, almost all of them are essentially the same in the following sense: they are obtained from ω-categorical theories, called base theories below, by adding axioms for infinitely many constant symbols. Moreover all known base theories have the (strict) order property in the sense of [6], and so they are unstable. For example, Ehrenfeucht's well-known example which has three countable models has the theory of dense linear order as its base theory. Many papers including [4] and [5] are motivated by the conjecture that every non-ω-categorical theory with a finite number of countable models has the (strict) order property, but this conjecture still remains open. (Of course there are partial positive solutions. For example, in [4], Pillay showed that if such a theory has few links (see [1]), then it has the strict order property.) In this paper we prove the instability of the base theory T0 of such a theory T rather that T itself. Our main theorem is a strengthening of the following which is also our result: if a theory T0 is stable and ω-categorical, then T0 cannot be extended to a theory T which has n countable models (1 < n < ω) by adding axioms for constant symbols.

In [3], Shelah defined the cardinals κn(T) and , for each theory T and n < ω. κn(T) is the least cardinal κ without a sequence (pi)i<κ of complete n-types such that pi is a forking extension of pj for all i < j < κ. It is essential in computing the stability spectrum of a stable theory. On the other hand is called the number of independent partitions of T. (See Definition 1.2 below.) Unfortunately this invariant has not been investigated deeply. In the author's opinion, this unfortunate situation of is partially due to the fact that its definition is complicated in expression. In this paper, we shall give equivalents of which can be easily handled. In §1 we shall state the definitions of κn(T) and . Some basic properties of forking will be stated in this section. We shall also show that if = ∞ then T has the independence property. In §2 we shall give some conditions on κ, n, and T which are equivalent to the statement . (See Theorem 2.1 below.) We shall show that does not depend on n. We introduce the cardinal ı(T), which is essential in computing the number of types over a set which is independent over some set, and show that ı(T) is closely related to . (See Theorems 2.5 and 2.6 below.) The author expects the reader will discover the importance of via these theorems. Some of our results are motivated by exercises and questions in [3, Chapter III, §7]. The author wishes to express his heartfelt thanks to the referee for a number of helpful suggestions.