= Solution
Write $f_n(q)=\mathbb P(G(n,q)\in P_n)$. This is increasing in $q$ by <monotone coupling of binomial random graphs>, and $f_n(p)=1/10$ by hypothesis.
First let $q/p\to0$. Choose $k=\lfloor p/(2q)\rfloor$, so $k\to\infty$. The union of $k$ independent copies of $G(n,q)$ has distribution $G(n,q')$, where
$$
q'=1-(1-q)^k\leq kq\leq p.
$$
If any layer has $P_n$, their union has $P_n$, and hence
$$
\frac1{10}=f_n(p)\geq f_n(q')
\geq1-(1-f_n(q))^k.
$$
It follows that $f_n(q)\leq1-(9/10)^{1/k}\to0$.
Now let $q/p\to\infty$. Choose $k=\lfloor q/(2p)\rfloor$, again tending to infinity. The union of $k$ independent $G(n,p)$ graphs has parameter
$$
p'=1-(1-p)^k\leq kp\leq q.
$$
Monotonicity and independence give
$$
f_n(q)\geq f_n(p')
\geq1-(1-f_n(p))^k
=1-(9/10)^k\longrightarrow1.
$$
Thus $p(n)$ is a threshold function.
Back to article page
