Homogeneous 1‐based structures and interpretability in random structures

Let V be a finite relational vocabulary in which no symbol has arity greater than 2. Let M be countable V ‐structure which is homogeneous, simple and 1‐based. The first 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 first, implies (without the assumption on primitivity) that if M is “coordinatized” by a set with SU‐rank 1 and there is no definable (without parameters) nontrivial equivalence relation on M with only finite classes, then M is strongly interpretable in a random structure.