SOME POSITIVE RESULTS IN THE CONTEXT OF UNIVERSAL MODELS 5 : (4) if Γn(x) is consistent with Tn, then Γn(c) ⊆ Tn+1 for some c ∈ L∗ \ L. The induction is straightforward and we obtain that T ∗ = nω Tn is a maximal consistent theory in L∗. Let M be a countable model of it, therefore by (3), M = (M, an)nω where M = {an : n ω}. M is clearly a model of T and (4) guarantees that it is saturated. 2.8 Of course, saturation is not a neccessary condition for universality: ω + Q is a universal countable linear order but is not a saturated model. We shall not develop this topic further in this article and from now on we shall only consider uncountable models. 3. Changing the cardinal arithmetic When we leave the realm of GCH and its remnants we are more or less left with universes which we construct with forcing and where instances of GCH are violated by the construction of the extension. Theorem 2.7(2) shows that if we are not careful about how we do this, we shall end up basically with no universal models of any sort. Theorem 2.7(3) shows that for certain theories such as linear orders, no matter how careful we are, if we violate GCH suﬃciently (including making 2κ = 2κ+), the universality number at κ+ will jump to the largest possible value of 2κ+. We shall see below that for certain other theories, for example theory of graphs, it is possible to violate GCH as much as we like and still keep the universality number low. This indicates that the ability of having a small universal number in ‘reasonable’ forcing extensions in which the relevant instances of GCH are violated is a property of the theory itself, which is not possessed by all theories. In fact, in a series of papers, e.g. [7], [8], [6], [15], [16], [3] the thesis claiming the connection between the complexity of a theory and its amenability to the existence of universal models, has been pursued. In [4] we introduced the following definition, which formalised these notions: Definition 3.1. We say that a theory T is amenable iff whenever λ is an uncountable cardinal satisfying λλ = λ and 2λ = λ+, while θ satisfies cf(θ) λ+, there is a λ+-cc ( λ)-closed forcing notion that forces 2λ to be θ and the universality number of T at λ+ to be smaller than θ. Localising this definition at a particular λ we define what is meant by theories that are amenable at λ. Connected to this definition there is a somewhat technical definition of high non-amenability (see Definition 0.3 of [4]). We shall not quote the definition but state only that high non-amenability of T implies that T is not amenable in the sense of Definition 3.1, and that the theory of dense linear order with no endpoints is a prototypical example of a highly non-amenable T . We should also comment that the amenability/high non-amenability is envisioned as a dividing in the classification theory of unstable theories. The exact syntactic properties that correspond to this line have not been found yet, but is known that the property SOP4 implies high non- amenability and simplicity implies amenability. We shall not go into these model- theoretic considerations at this point but refer the reader to the articles mentioned above. Here we shall simply be concerned to give examples and techniques which apply to amenable theories. For simplicity in presentation we shall concentrate on the case of amenability at λ = ℵ1 but we warn the reader that there are some caveats in looking only at this case-we shall indicate them below.

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