Let $X$ denote the number of triangles in a random graph $G$ chosen from $G(n, p)$. Find the mean and variance of $X$. Hence show that $p=n^{-1}$ is a threshold for the existence of a triangle, in the sense that if $p n \rightarrow 0$ then almost surely $G$ does not contain a triangle, while if $p n \rightarrow \infty$ then almost surely $G$ does contain a triangle.

Now let $p=n^{-1 / 2}$, and let $Y$ denote the number of edges of $G$ (chosen as before from $G(n, p))$. By considering the mean of $Y-X$, show that for each $n \geqslant 3$ there exists a graph on $n$ vertices with at least $\frac{1}{6} n^{3 / 2}$ edges that is triangle-free. Is this within a constant factor of the best-possible answer (meaning the greatest number of edges that a triangle-free graph on $n$ vertices can have)?