Let $L$ be a first-order language, let $M$ be an $L$-structure, and let $\kappa$ be an infinite cardinal. Then $M$ is $\kappa$-saturated if and only if the following condition holds: for every subset $A \subset M$ with $|A| < \kappa$, every finite tuple of variables $x = (x_1,\dots,x_n)$ with $n \geq 1$, and every partial $L(A)$-type $\Sigma(x)$ finitely satisfiable in $M$, there exists a tuple $a \in M^n$ such that \begin{align*} M \models \varphi(a) \end{align*} for every formula $\varphi(x) \in \Sigma(x)$.

Let $T$ be a complete theory in a language $L$, let $M\models T$, and let $\kappa$ be an infinite cardinal such that $|L|<\kappa$, $\kappa^{<\kappa}=\kappa$, and $|M|\le \kappa$. Then there is an elementary extension $N\succeq M$ such that $|N|\le \kappa$ and $N$ realizes every complete type in finitely many variables over every parameter set $A \subset N$ with $|A| < \kappa$.

Let $T$ be a complete countable stable theory, let $A$ be a small parameter set, and let $D$ be an $A$-definable strongly minimal set. Assume that for every independent set $I \subset D$ there is a model $M_I$ prime and atomic over $A \cup I$, and that finite independent tuples from $D$ of the same length have the same type over $A$. If $I,J \subset D$ are independent sets with $|I|=|J|$, then $M_I \cong_A M_J$.

For a complete theory $T$, the following are equivalent: 1. $T$ is stable in some infinite cardinal $\kappa$ with $\kappa \ge |T|$. 2. For every infinite cardinal $\kappa$ satisfying $\kappa^{|T|} = \kappa$, the theory $T$ is stable in $\kappa$. 3. No formula has the order property with respect to $T$.

Let $T$ be a complete first-order theory in a countable language. Suppose there is a formula $\varphi(x;y)$ of $T$, where $x$ and $y$ are finite tuples of variables, with the order property: for every $n \in \mathbb{N}$ there exist tuples $a_1,\dots,a_n$ and $b_1,\dots,b_n$ in some model of $T$ such that \begin{align*} \models \varphi(a_i;b_j) \quad \Longleftrightarrow \quad i < j \end{align*} for all $1 \le i,j \le n$. Then for every uncountable cardinal $\kappa$, there exist two non-isomorphic models $M,N \models T$ with $|M| = |N| = \kappa$.

Let $D$ be a parameter-free definable strongly minimal set in a monster model $\mathfrak{M}$ of a complete theory. For $A \subset D$, define \begin{align*} \operatorname{cl}: \mathcal{P}(D) \to \mathcal{P}(D), \qquad \operatorname{cl}(A)=D \cap \operatorname{acl}_{\mathfrak{M}}(A). \end{align*} Then $\operatorname{cl}$ is a pregeometry on $D$: it is extensive, increasing, idempotent, has finite character, and satisfies exchange.

Let $T$ be strongly minimal and let $M\models T$. For every $A\subseteq M$, every set $I\subseteq M$ that is independent over $A$ extends to a basis of $M$ over $A$, and any two bases of $M$ over $A$ have the same cardinality.

Let $L$ be a first-order language, let $M$ be an $L$-structure, let $A \subseteq M$ be a parameter set, and let $n \in \mathbb{N}$. Let $S_n(A)$ denote the Stone space of complete $n$-types over $A$ in variables $\bar{x} = (x_1,\dots,x_n)$, equipped with the Stone topology whose basic open sets are \begin{align*} [\theta] := \{p \in S_n(A) : \theta(\bar{x}) \in p\}, \end{align*} where $\theta(\bar{x})$ ranges over all $L(A)$-formulas in the variables $\bar{x}$. For every $L(A)$-formula $\varphi(\bar{x})$, the subset $[\varphi] \subseteq S_n(A)$ is both open and closed. Moreover, for all $L(A)$-formulas $\varphi(\bar{x})$ and $\psi(\bar{x})$, \begin{align*} [\neg \varphi] &= S_n(A) \setminus [\varphi], \\ [\varphi \wedge \psi] &= [\varphi] \cap [\psi], \\ [\varphi \vee \psi] &= [\varphi] \cup [\psi]. \end{align*}…

Let $L$ be a first-order language, and let $T$ be a complete $L$-theory with quantifier elimination. Let $M,N \models T$. Let $A$ be a parameter set equipped with interpretations $i_M:A \to M$ and $i_N:A \to N$ whose named constants have matching quantifier-free diagram in $M$ and $N$; for example, $A$ may be a common substructure of both models with the same interpretation of all $L$-terms. If tuples $b \in M^n$ and $c \in N^n$ have the same quantifier-free type over $A$ via the interpretations $i_M$ and $i_N$, then they have the same complete type over $A$ via those interpretations.

Let $L$ be a first-order language, let $T$ be a complete $L$-theory, and let $M$ and $N$ be countable atomic models of $T$. Then $M \cong N$ as $L$-structures. In particular, if $L$ is countable, then any two countable prime models of $T$ are isomorphic, since countable prime models of a complete theory in a countable language are atomic.