The random graph is the Fraisse limit of the class of finite graphs, the random directed graph is the Fraisse limit of the class of directed graphs, a directed graph is just a set with a binary relation.

It’s easy to see that the random directed graph interprets the random graph, in fact the second is a reduct of the first. I am curious to know about the other direction. I don’t see an easy way to do it, or an easy way to rule it out. More generally, I would be curious have pointers to research on when countable homogeneous structures interpret other countable homogeneous structures. I don’t know if anyone has thought about this, I’m not very familiar with that terrain.

Motivation: I am doing some work with a weak notion of interpretability. I show that the random graph weakly interprets the random directed graph, and I use this as a fairly important lemma. So I’m curious to know if anything is known/obvious to experts about actual interpretations. If there is an actual interpretation then it would be a bit silly to spend a page constructing a weak interpretation.

I am actually thinking about the random $k$-ary hypergraph and the random $k$-ary relation.