Joint embedding property

Abstract: If T is a (first order) universal theory then we can consider the category of existentially closed models of T with embeddings as morphisms. This class of structures is in general not the class of models of a first order theory T ′. (It will be, just if T has a model companion T ′.) The “model theory” of such a “non-elementary class” was a substantial line of research in the 1970’s, mainly among the group of model-theorists around Abraham Robinson. In [10], Shelah developed some aspects of stability theory for such a category, concentrating on counting (existential) types. He asked implicitly whether the theory of forking can be developed in such a context. We give a positive solution in this paper, developing the theory of forking for the category of existentially closed models of an arbitrary universal theory T , assuming a suitable notion of “simplicity”. Under the additional assumption that T has the amalgamation property (AP) and joint embedding property (JEP), Shelah did develop forking in some form. Hrushovski ([4]) rediscovered this latter class of theories (universal T with AP and JEP) calling them Robinson theories, and pointing out that all model-theoretic methods should apply to the category of e.c. models of such T . For Robinson theories, quantifier-free types are the main object of study, and as quantifier-free formulas are closed ∗Supported by NSF grant DMS-96-96268

Abstract: We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number $\kappa$, approximately 2.20557, for which there are only countably many permutation classes of growth rate (Stanley-Wilf limit) less than $\kappa$ but uncountably many permutation classes of growth rate $\kappa$, answering a question of Klazar. We go on to completely characterize the possible sub-$\kappa$ growth rates of permutation classes, answering a question of Kaiser and Klazar. Central to our proofs are the concepts of generalized grid classes (introduced herein), partial well-order, and atomicity (also known as the joint embedding property).

Abstract: This article is concerned with classes of relational structures that are closed under taking substructures and isomorphism, that have the joint embedding property, and that furthermore have the Ramsey property, a strong combinatorial property which resembles the statement of Ramsey's classic theorem Such classes of structures have been called Ramsey classes Nesetril and Roedl showed that they have the amalgamation property, and therefore each such class has a homogeneous Fraisse-limit Ramsey classes have recently attracted attention due to a surprising link with the notion of extreme amenability from topological dynamics Other applications of Ramsey classes include reduct classification of homogeneous structures We give a survey of the various fundamental Ramsey classes and their (often tricky) combinatorial proofs, and about various methods to derive new Ramsey classes from known Ramsey classes Finally, we state open problems related to a potential classification of Ramsey classes

Abstract: It is consistent with 'CH that every universal theory of relational structures with the joint embedding property and amalgamation for Ρ − (3)-diagrams has a universal model of cardinality N 1 . For classes with amalgamation for Ρ − (4)-diagrams it is consistent that 2 No >N 2 and there is a universal model of cardinality N 2