On the maximal number of elements pairwise generating the finite alternating group

Let $G$ be the alternating 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, when $n$ varies in the family of composite numbers, $\sigma(G)/\omega(G)$ tends to $1$ as $n \to \infty$. Moreover, we explicitly calculate $\sigma(A_n)$ for $n \geq 21$ congruent to $3$ modulo $18$.