= Solution
For a <first-order theory> $T$ in a <first-order language> $L$, a complete $n$-type is a maximal $T$-consistent set $p(x_1,\ldots,x_n)$ of <first-order formula>[formulas] whose free variables lie among $x_1,\ldots,x_n$. Equivalently, it chooses exactly one of $\varphi$ and $\neg\varphi$ for every such formula while remaining consistent with $T$.
An <isolated type> $p$ is isolated by a formula $\theta(\bar x)$ when $T\cup\{\exists\bar x\,\theta(\bar x)\}$ is consistent and
$$
T\models\forall\bar x\,(\theta(\bar x)\to\varphi(\bar x))
$$
for every $\varphi\in p$. The type is an <omitted type> in an $L$-structure $M$ when no tuple $\bar a\in M^n$ satisfies every formula in $p$.
The <omitting types theorem> states that if $T$ is a consistent theory in a countable language and $(p_i)_{i\in\mathbb N}$ is a countable family of nonisolated finite-arity types, then $T$ has a countable model omitting every $p_i$.
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