Independence and the finite submodel property

We study a class C of ℵ 0 -categorical simple structures such that every M in C has uncomplicated forking behavior and such that definable relations in M which do not cause forking are independent in a sense that is made precise; we call structures in C independent. The SU-rank of such M may be n for any natural number n > 0 . The most well-known unstable member of C is the random graph, which has SU-rank one. The main result is that for every strongly independent structure M in C , if a sentence φ is true in M then φ is true in a finite substructure of M . The same conclusion holds for every structure in C with SU-rank one; so in this case the word ‘strongly’ can be removed. A probability theoretic argument is involved and it requires sufficient independence between relations which do not cause forking. A stable structure M belongs to C if and only if it is ℵ 0 -categorical, ℵ 0 -stable and every definable strictly minimal subset of M eq is indiscernible.