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

Let G be the alternating group of degree n. Let $\omega \left(G\right)$ be the maximal size of a subset S of G such that $〈x,y〉=G$ whenever $x,y\in S$ and $x\ne y$ and let $\sigma \left(G\right)$ 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, $\sigma \left(G\right)/\omega \left(G\right)$ tends to 1 as $n\to \infty$. Moreover, we explicitly calculate $\sigma \left(A_n\right)$ for $n\ge 21$ congruent to 3 modulo 18.