Thanks for the super quick answer! I just want to check, what is $e_v$? For $v\in {\bf F}_3^n$ is it the function given by $e_v(x) = 1$ if $x=v$ and $e_v(x) = 0$ otherwise?

@Seva Yes, sorry that I am not being very clear. I do actually mean to say that no two triangles in the graph share an edge. Furthermore, one can assume that every edge is in a triangle. If you have an answer, I would be happy to accept it, since my question is too general to really be interesting but at least for this special case we could get a bound.

@Seva Yes, the graph is a union of triangles that possibly share vertices but not edges.

Okay, what went wrong was my statement that if the edge set of the graph is a union of disjoint triangles, then the number of cherries is the number of edges. This is wrong because the triangles may share vertices (but not edges).

@Seva Thanks for this calculation! It is quite surprising to me that the sum of the squares is on the same order as the number of edges in my tripartite graph because it raises an apparent contradiction. I will work it out precisely to see what is going on.

@Seva In the case where the edge set is a union of disjoint triangles, the number of cherries is exactly the number of edges. Does this mean that $d(G)$ is roughly the number of edges?

@JukkaKohonen I will post the arithmetic progressions question separately! Indeed, even though the poset is not graded there are still lots of questions to ask about it, and I'm not sure if anything interesting will pop up, but I will continue looking into it for sure!

Oops, that's embarrassing. I guess that sort of makes the question senseless

Thanks for this answer! I realise now that the problem was too broad and have edited my original post to include my motivating example.