2 MIRNA DZAMONJAˇ The article is organised as follows: it first presents some classical results from model theory that apply in ZFC or under the GCH-like assumptions. This is the content of §2, which is divided to subsections relating to saturated and special models and countable universal models. Section 3 moves into the realm where CH is violated and considers the possible existence of universal models in forcing exten- sions where CH fails, concentrating on ℵ1. This section is divided into subsections dealing with graphs, triangle-free graphs, linear orders and amenability at ℵ1. Throughout κ stands for an infinite cardinal. An unattributed T stands for a theory, which means a complete first order theory with infinite models. For simplicity in this presentation we restrict ourselves to the case of countable theories. A type for us is any consistent set of sentences, and a complete type is a maximal consistent set of sentences. By a universal model of T of size κ we mean a model in which every other model of T of size κ embeds elementarily1. 2. Some classical results We present some results on the existence of certain kinds of universal models for complete first order theories, again restricting to the case of countable theories. Results presented in this section mostly come from 1960s. 2.1. Saturated and special models. Definition 2.1. A model M of a theory T is said to be κ-saturated if for every A ⊆ M of size κ, the expansion (M, a)a∈A realises every type Γ(x) of the expanded language which is consistent with the complete theory Th(M, a)a∈A. M is said to be saturated if it is |M|-saturated. A generalisation of Cantor’s proof that the rationals are a unique countable dense linear order with no first or last elements, gives us that saturated models are universal. See Theorem 2.4 for a detailed statement. The basic theorem about the existence of saturated models at uncountable cardinals is the following Lemma 2.2. (Vaught, [10]) Suppose that N is a model T of size ≤ 2κ. Then there is a κ+-saturated extension M of N of size 2κ. Proof. Note that |[N]κ| = 2κ, and for every A ∈ [N]κ the language LA = L∪{ca}a∈A has size κ, therefore the total number of relevant types is 2κ. Introduce for each such type Σ a new symbol cΣ. We can form the set of sentences Γ consisting of the elementary diagram of N along with Σ(cΣ) for all relevant Σ. This is a finitely satisfiable set of sentences, hence it has a model, so it has a model of size 2κ. Let M be the reduction of this model to the original language. 2.2 Provided that we assume some cardinal arithmetic this now gives us the exis- tence of saturated models in successor cardinals: Theorem 2.3. (Vaught, [10]) Suppose that κ satisfies 2κ = κ+. Then there is a saturated model M of T of size κ+. Proof. Let N be any model of T of size 2κ. By induction on α 2κ we choose models Nα of T so that • N0 = N, Nα ≺ Nα+1, Nδ = αδ Nα for δ limit 0, 1 In this context, because of the compactness theorem, we could equivalently require that every model of infinite size ≤ κ embeds into the κ-universal model.

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