= Solution
The <Ryll-Nardzewski theorem> says that for a <complete theory> $T$ in a <countable> <first-order language> with infinite <first-order models>, the following are equivalent: $T$ is $\aleph_0$-categorical; for every positive <integer> $n$, the <set> $S_n(T)$ of complete parameter-free $n$-types is finite; every such finite-arity <complete type> is isolated by a <logical formula>. The countable-language hypothesis and the requirement for every finite arity are important.
The <omitting types theorem> says that, for a <first-order consistent> <complete theory> in a <countable> <first-order language>, a <countable> family of <nonprincipal partial types> in finite <tuples> of <logical variables> can all be omitted in a <countable> <first-order model>. Here nonprincipal means that no <logical formula> <first-order consistent> with the <first-order theory> implies every <logical formula> of the <partial type>. Omission means that no <tuple> in the constructed <first-order model> satisfies all <logical formulas> of that <partial type>. The <complete one-type> version is the special case of one <partial type> in one <logical variable>.
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