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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.