Probabilistic generation of finite simple groups, II

In earlier work it was shown that each nonabelian finite simple group G has a conjugacy class C such that, whenever 1≠x G, the probability is greater than 1/10 that G= x,y for a random y C. Much stronger asymptotic results were also proved. Here we show that, allowing equality, the bound 1/10 can be replaced by 13/42; and, excluding an explicitly listed set of simple groups, the bound 2/3 holds. We use these results to show that any nonabelian finite simple group G has a conjugacy class C such that, if x1, x2 are nontrivial elements of G, then there exists y C such that G= x1,y = x2,y . Similarly, aside from one infinite family and a small, explicit finite set of simple groups, G has a conjugacy class C such that, if x1, x2, x3 are nontrivial elements of G, then there exists y C such that G= x1,y = x2,y = x3,y . We also prove analogous but weaker results for almost simple groups.