Let $\Lambda$ be the ring of symmetric functions over $\mathbb{Z}$ equipped with the Hall [inner product](/page/Inner%20Product) for which the Schur functions $\{s_\kappa\}$ form an [orthonormal basis](/page/Orthonormal%20Basis). For each partition $\lambda$, let $s_\lambda^\perp: \Lambda \to \Lambda$ denote the linear operator adjoint to multiplication by $s_\lambda$, so that \begin{align*} \langle s_\lambda^\perp f, g\rangle = \langle f, s_\lambda g\rangle \end{align*} for all $f,g \in \Lambda$. Then for all partitions $\lambda,\mu,\nu$, \begin{align*} s_\lambda^\perp(s_\mu s_\nu) = \sum_{\alpha,\beta} c_{\alpha\beta}^{\lambda} \bigl(s_\alpha^\perp s_\mu\bigr) \bigl(s_\beta^\perp s_\nu\bigr), \end{align*} where $c_{\alpha\beta}^{\lambda}$ denotes the Littlewood-Richardson coefficient…

Shelah's main gap says, informally, that complete countable first-order theories split into a structure side and a nonstructure side. On the structure side, the classifiable theories have models in uncountable cardinals described by invariants arising from stable decomposition. On the nonstructure side, the non-classifiable theories satisfy precise spectrum theorems giving the maximum possible number of pairwise non-isomorphic models in the relevant uncountable cardinals.

Let $T$ be a complete countable first-order theory, and suppose that $T$ is categorical in some uncountable cardinal. Let $\mathfrak C$ be a monster model of $T$. Then there exist a finite tuple $a \in \mathfrak C$ and a formula $\varphi(x,a)$ such that the definable set \begin{align*} D := \varphi(\mathfrak C,a) \end{align*} is strongly minimal in the expansion $\mathfrak C_a$ by constants for $a$. Moreover, if $p$ is any non-algebraic regular type of $T(a)$, then $p$ is nonorthogonal to the generic type of the strongly minimal set $D$.

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 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 $L$ 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.

Let $T$ be stable, let $\mathfrak C \models T$ be a monster model, let $M \preceq \mathfrak C$ be $\kappa$-saturated, and let $B=M\cup A\subseteq\mathfrak C$ with $|A|<\kappa$. Every type $p(x)\in S_x(M)$ has an extension $q(x)\in S_x(B)$ that is the restriction of a global $M$-invariant type built from an $M$-definitional scheme for $p$.

Let $L$ be a countable first-order language, and let $T$ be a complete $L$-theory with infinite models. For each $n \geq 1$, let $S_n(T)$ denote the Stone space of complete $n$-types over $\varnothing$ consistent with $T$. The following conditions are equivalent: 1. $T$ is $\omega$-categorical, meaning that any two countable models of $T$ are isomorphic. 2. For every $n \geq 1$, the Stone space $S_n(T)$ is finite. 3. For every $n \geq 1$, there are only finitely many $L$-formulas with free variables among $x_1,\dots,x_n$, up to equivalence modulo $T$. 4. For some countable model $M \models T$, the automorphism group $\operatorname{Aut}(M)$ is oligomorphic, meaning that for every $n \geq 1$, the componentwise action of $\operatorname{Aut}(M)$ on $M^n$ has only finitely many orbits.

Let $T$ be a complete first-order theory in a language $\mathcal{L}$. If $M \models T$ is countable and atomic, meaning that for every finite tuple $a \in M^n$ the complete type $\operatorname{tp}^M(a/\varnothing)$ is isolated by an $\mathcal{L}$-formula, then $M$ is a prime model of $T$: for every model $N \models T$, there exists an elementary embedding $f: M \to N$.

Let $T$ be a complete first-order theory in a language $\mathcal{L}$, let $n \in \mathbb{N}$, and let $p(x) \in S_n(T)$ be a complete $n$-type over $T$, where $x = (x_1,\dots,x_n)$. Suppose that an $\mathcal{L}$-formula $\theta(x)$ isolates $p(x)$, in the sense that for every $\mathcal{L}$-formula $\varphi(x)$, \begin{align*} \varphi(x) \in p(x) \implies T \models \forall x\,(\theta(x) \to \varphi(x)). \end{align*} If \begin{align*} T \models \exists x\,\theta(x), \end{align*} then every model $M \models T$ realizes $p(x)$. Equivalently, for every $M \models T$, there exists $a \in M^n$ such that \begin{align*} M \models \varphi(a) \end{align*} for every $\varphi(x) \in p(x)$.