Solution
= Solution
Only triangle pairs sharing an edge have nonzero covariance, so
$$
\operatorname{var}T=\binom n3(p^3-p^6)+2\binom n2\binom{n-2}2(p^5-p^6).
$$
After division by $(\mathbb ET)^2$ this is $O(n^{-3}p^{-3}+n^{-2}p^{-1})\to0$ for $p=n^{-\alpha}$, $0\lt \alpha\lt 1$; use part (iii).