Someone can to help me in the following problem about the threshold function of the property Sperner set? I don't know where to start.
Let $\mathcal{F} \subseteq \mathcal{P}([n])$ be a random family of subsets with $\mathbf{Pr}(A\in \mathcal{F}) = p(n)$, for all $A\subseteq [n]$ (independently). Show that $p_0(n) = 3^{-n/2}$ is a threshold function for the event "$\mathcal{F}$ is Sperner", i.e., $$ \mathbf{Pr}(\mathcal{F} \text{ is Sperner}) \xrightarrow{n\to\infty} \left\{ \begin{array}{ll} 0, & \text{if }\; p(n) \ll p_0(n) \hbox{;} \\ 1, & \text{if }\; p(n) \gg p_0(n)\hbox{.} \end{array} \right. $$
Note: $\mathcal{F}$ is Sperner if $\forall A,B \in \mathcal{F}$, we have $A \not\subseteq B$ and $B \not\subseteq A$.