Abstract

We study a class of 0-categorical simple structures such that every M in 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 independent. The SU-rank of such M may be n for any natural number n>0. The most well-known unstable member of is the random graph, which has SU-rank one. The main result is that for every strongly independent structure M in , 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 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 if and only if it is 0-categorical, 0-stable and every definable strictly minimal subset of is indiscernible