(I am skipping any setup stuff and speaking roughly)
One fact that I am sure of is that a definable subset $X$ is fixed by all automorphisms of the (super)structure.
I simply wonder the converse:
"If a subset $X$ is fixed by all automorphisms, then $X$ is a definable set."
Is this true (in general)? If not, could you provide some counterexample and the cases/conditions that the latter assertion is true?
On
Of course this is true if your structure is big enough (sufficiently saturated), which is kind of assumed by putting this question into model theory. In-fact the following statement holds: A set $D$ is definable over $A$ iff it is fixed by all automorphism fixing $A$. to proof right to left note that the $D$-membership depends only on the type over $A$. Then check that $\{tp(d/D):d\in D\}=X$ is a clopen set in the type space to finish the proof (Check that $X$ and $S(A)-X$ are closed).
Look at the naturals under the usual addition, multiplication. The only automorphism is the trivial one, but there are many undefinable subsets.
The same is true of the reals.