Automorphisms vs color-preserving automorphisms in Fraisse classes

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 \cal K of finite structures over a finite relational vocabulary, are the following two conditions equivalent? Below, by a colored structure from \cal K we mean a structure from \cal K 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 \cal K have boundedly many automorphisms: for some number n, every structure in \cal K has at most n automorphisms.

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

One thought on “Automorphisms vs color-preserving automorphisms in Fraisse classes”

  1. 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.

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