**Question:** If the question is about proving $P$, which, pointed out in comments, follows from $Q$ and $R$. Is it acceptable to write an answer only containing the proof of $Q$, as part of collaboration?

This is what I have encountered in this question. I proposed a random graph model, and what remains is to show that

- $Q$: It has girth 5, and
- $R$: It has unbounded chromatic number.

I can prove the graph has girth 5 with high probablity under my model, but I'm not sure whether they have unbounded chromatic number. Maybe someone can prove $R$, and it would complete the answer.