Solution
= Solution
With $p=c\log n/n$, the inequality $1-p\leq e^{-p}$ gives
$$
\mathbb E(N)
=n(1-p)^{n-1}
\leq ne^{-p(n-1)}
=n^{\,1-c(n-1)/n}
\longrightarrow0
$$
when $c>1$. The <Markov inequality> now gives
$$
\mathbb P(N>0)\leq\mathbb E(N)\longrightarrow0.
$$
Consequently the upper side of the <isolated-vertex threshold in the Erdős-Rényi model> is
$$
\boxed{\mathbb P(N=0)\longrightarrow1}.
$$