Let $G(n,p)$ be the binomial random graph with vertex set $[n]$. Let $E_k$ denote the event that $G(n,p)$ contains some collection of $k$ edge-disjoint triangles. Show that $\mathbb{P}[E_k] \leq \frac{\left(\binom{n}{3}p^3\right)^k}{k!}$.
This seems like the number of edge-disjoint triangles follows a Poisson paradigm. Maybe we can prove it using the toolkits of Janson's inequalities, but I'm not sure how to.