Homogeneous 1-based structures and interpretability in random structures

Let V be a nite relational vocabulary in which no symbol has arity greater than 2. Let M be countable V -structure which is homogeneous, simple and 1based. The rst main result says that if M is, in addition, primitive, then it is strongly interpretable in a random structure. The second main result, which generalizes the rst, implies (without the assumption on primitivity) that if M is coordinatized by a set with SU-rank 1 and there is no de nable (without parameters) nontrivial equivalence relation on M with only nite classes, then M is strongly interpretable in a random structure.