Some positive results in the context of universal models Mirna Dˇzamonja Abstract. Let (K, ≤) be a quasi-ordered set or a class, which we think of as a class of models. A universal family in K is a dominating family in (K, ≤), and if there is such a family of size one then we call its single element a universal model in K. We survey some important instances of the existence of small universal families and universal models in various classes and point out the influence of the axioms of set theory on the existence of such objects. Then we present some of the known methods of constructing small universal families and universal models and discuss their limitations, pointing out some of the remaining open questions. 1. Introduction Let (K, ≤) be a a quasi-ordered set or a class, which we think of as a class of models. In the context that interests us this may be the class of models of a given cardinality of some first order theory ordered by elementary embedding or the class of models of a given cardinality of some abstract elementary class quasi-ordered by the inherited order. We may also consider classes whose membership is not determined by cardinality but by some other cardinal invariant such as topological weight. A universal family in K is a dominating family in (K, ≤), and if there is such a family of size one then we call its single element a universal model in K. The smallest size of a universal family is called the universality number of (K, ≤). Immediate examples of universal models are the the rationals considered as a linear order, which embed every countable linear order, or [0, 1]κ which contains a closed copy of every compact space of weight κ, or the random graph which embeds every countable graph. There are many other examples in just about every branch of mathematics. The purpose of this article is to discuss general methods which can be used to demonstrate the existence of universal models in various specific contexts. In this presentation we concentrate on countable first order theories. The article does not deal with the related subject of methods that can be used to demonstrate that a certain theory does not have a small universal family at a certain cardinal we can refer the reader to the survey article [2] for a description of some such ideas. 1991 Mathematics Subject Classification. Primary 03E35, 03C55. Key words and phrases. universal models, universal families. The author thanks Mittag-Leffler Institute for their support in September 2009 and EPSRC for their support through grant EP/G068720. 1 Contemporary Mathematics Volume 533, 2010 c 2010 American Mathematical Society 1 http://dx.doi.org/10.1090/conm/533/10501

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