= Finite invariant sets in a countable saturated structure are definable
Let $M$ be a countable <omega-saturated model> and $D\subseteq M$ finite. Then $D$ is a parameter-free <definable set> exactly when every <structure automorphism> preserves it setwise. One direction is <automorphism invariance of parameter-free definability>. For the converse, <saturation implies homogeneity> and the <back-and-forth method> show that equal empty-set types are in the same automorphism orbit. For $a\in D$, if no formula in $\operatorname{tp}(a)$ has its realization set contained in $D$, the partial type $\operatorname{tp}(a)\cup\{x\ne d:d\in D\}$ is finitely satisfiable. Its parameters form a finite set, so saturation realizes it outside $D$, contradicting invariance. Choose a defining neighbourhood for each $a\in D$ and take their finite disjunction. The empty set is defined by $x\ne x$. Setwise invariance permits automorphisms to permute points within $D$.
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