Pairwise generating and covering sporadic simple groups

Let G be a non-cyclic finite group that can be generated by two elements. A subset S of G is said to be a pairwise generating set for G if every distinct pair of elements in S generates G. The maximal size of a pairwise generating set for G is denoted by ω(G). The minimal number of proper subgroups of G whose union is G is denoted by σ(G). This is an upper bound for ω(G). In this paper we give lower bounds for ω(G) and upper bounds for σ(G) whenever G is a sporadic simple group.