Let G be the random graph $G\left(n,\frac{\log(n)}n\right)$, i.e. the graph on $n$ vertices with each edge independently having probability $\frac{\log(n)}n$ of being in the graph.
I wrote a program to find the number of triangles in this graph, call it $X$. Intuitively and from my results it seems that $E_n[X]$ tends to 0 as $n\rightarrow \infty$. I am wondering how I might go about proving this. For $G(n,c/n)$ I know that $E_n[X]=\frac{c^3}{6}$ as $n \rightarrow \infty$, not sure if this could be relevant.