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.
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,
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  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.
Volume 533, 2010
c 2010 American Mathematical Society
Volume 533 , 2011
c 2011 American Mathematical Society