Again, a question closely related to a characterization of standard atoms (not to be confused with standard alphabets:). The question is a refinement of a question posted in Automorphisms vs color-preserving automorphisms in Fraisse classes, which has been answered negatively (see the recent post The colored open question closed).
Consider a Fraisse class of finite structures over a finite relational vocabulary. 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.
A structure from is stiffened by a subset of its carrier if the subgroup of automorphisms
is trivial, i.e. contains only the identity. Similarly one defines when a colored structure is stiffened by , by considering color-preserving automorphisms of instead of all automorphisms.
Fact: The following conditions are equivalent, for every Fraisse class :
1. The colored structures in have boundedly many automorphisms: for every there is an such that every colored structure in with color classes of size bounded by , has at most color-preserving automorphisms.
2. The colored structures in are stiffened by a bounded number of elements: for every there is an such that every colored structure in with color classes of size bounded by , is stiffened by some its subset of at most elements.
Proof (idea): 1 implies 2: A set may be found in a greedy way, by inspecting elements of is an arbitrary order and checking whether imposing the equality eliminates some automorphisms of .
2 implies 1: If stiffens then every (colored) automorphism of is uniquely determined by its restriction to . As the color classes are bounded and is bounded, there is only boundedly many possibilites for the value of , and thus boundedly many color-preserving automorphisms.
Remark: It seems that the above condition is quite robust, as it may be formulated in a number of equivalent ways. For instance, one can define an independence system on color classes, and require a bound on the size of independent sets.
The conditions in the above fact may be adapted to a non-colored settings, by omiting the bound on the size of color classes. In case of condition 1, the colored and non-colored versions are inequivalent, as shown in a recent post The colored open question closed. An interesting open question remains for condition 2:
Question: Are the following two conditions equivalent, for every Fraisse class ?
1. The colored structures in are stiffened by a bounded number of elements.
2. The structures in are stiffened by a bounded number of elements: there is an such that every structure in is stiffened by some its subset of at most elements.