  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 $\Sym(n)$ or an alternating group
$\Alt(n)$, $\gamma(G)\geq cn$. This improves results of the first two authors
who had earlier proved that $a\varphi(n)\leq\gamma(G)\leq 2n/3,$ for some
positive constant $a$, where $\varphi$ is the Euler totient function. Bounds
are also obtained for the maximum size $\kappa(G)$ of a set $X$ of conjugacy
classes of $G=\Sym(n)$ or $\Alt(n)$ such that any pair of elements from
distinct classes in $X$ generates $G$, namely $cn\leq \kappa(G)\leq 2n/3$.
