Description
Abstract: Let A be a structure with a relational language. We say that A has the finite submodel property iff each first-order formula true in A is also true in a finite substructure of A. One important example of such a structure is the Rado graph (a countably infinite random graph) R. The well-known proof of the fact that R has the finite submodel property is based on the fact that there is an automorphism invariant probability measure defined on the family of definable subsets of R.
We will survey some known as well as recently obtained results on how this proof can be adapted for certain homogeneous structures. We will discuss both measure-theoretic and model-theoretic challenges that one must handle in related investigations.
The main results are as follows:
- a new sufficient condition implying that a stable structure has the finite submodel property,
- a weak converse of the whole approach: for certain structures, the finite submodel property implies the existence of certain
automorphism invariant probability measures mu_r (depending on r \in \omega) defined on the family of r-ary definable relations