# lo.logic – does the random graph interpret the random directed graph

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.