Here is a question closely related to a characterization of *standard atoms* (not to be confused with standard alphabets:). It asks whether a bound on the number of automorphisms is equivalent to a bound on the numer of color-preserving automorphisms, in every Fraisse class of finite relational structures.

For a Fraisse class of finite structures over a finite relational vocabulary, are the following two conditions equivalent? Below, by a *colored structure* from we mean a structure from together with a mapping from its carrier to some finite set of colors.

A *color class* is the set of elements mapped to the same color.

1. The structures in have boundedly many automorphisms: for some number , every structure in has at most automorphisms.

2. The colored structures in have boundedly many automorphisms: for every number there is a number such that every colored structure in with color classes of size bounded by , has at most color-preserving automorphisms.

The following question seems vaguely related.

Are the following conditions equivalent for a Fraisse class

1. All its structures are rigid, i.e. have trivial automorphism groups

2. There is a definable linear order in the Fraisse limit.

The above equivalence holds for Ramsey classes. Any Ramsey class is a Fraisse class, but not the other way around. The equivalence for Ramsey classes is mentioned here. It seems highly nontrivial, and follows from the work of Kechris, Pestov, Tedorcevic.