  Let $G$ be the symmetric group of degree $n$. Let $\omega(G)$ be the maximal
size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y
\in S$ and $x \neq y$ and let $\sigma(G)$ be the minimal size of a family of
proper subgroups of $G$ whose union is $G$. We prove that both functions
$\sigma(G)$ and $\omega(G)$ are asymptotically equal to $\frac{1}{2}
\binom{n}{n/2}$ when $n$ is even. This, together with a result of S. Blackburn,
implies that $\sigma(G)/\omega(G)$ tends to $1$ as $n \to \infty$. Moreover, we
give a lower bound of $(1-o(1))n$ on $\omega(G)$ which is independent of the
classification of finite simple groups. We also calculate, for large enough
$n$, the clique number of the graph defined as follows: the vertices are the
elements of $G$ and two vertices $x,y$ are connected by an edge if $\langle x,y
\rangle \geq A_n$.
