Normal coverings and pairwise generation of finite alternating and symmetric groups

The normal covering number $\gamma (G)$ of a finite, non-cyclic group $G$ is the least number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We prove that there is a positive constant $c$ such that, for $G$ a symmetric group $\text{Sym}(n)$ or an alternating group $\text{Alt}(n)$, $\gamma (G)\ge cn$. This improves results of the first two authors who had earlier proved that $a\phi (n)\le \gamma (G)\le 2n\mathrm{/}3,$ for some positive constant $a$, where $\phi$ is the Euler totient function. Bounds are also obtained for the maximum size $\kappa (G)$ of a set $X$ of conjugacy classes of $G=\text{Sym}(n)$ or $\text{Alt}(n)$ such that any pair of elements from distinct classes in $X$ generates $G$, namely $cn\le \kappa (G)\le 2n\mathrm{/}3$.