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 $⟨x,y⟩=G$ whenever $x,y\in S$ and $x\mathrm{\ne }y$ 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, $\sigma (G)\mathrm{/}\omega (G)$ tends to $1$ as $n\to \mathrm{\infty }$. Moreover, we explicitly calculate $\sigma (A_n)$ for $n\ge 21$ congruent to $3$ modulo $18$.