An irredundant base of a group G acting faithfully on a finite set Ξ
is a sequence of points in Ξ that produces a strictly descending chain
of pointwise stabiliser subgroups in G, terminating at the trivial subgroup.
Suppose that G is Snβ or Anβ acting
primitively on Ξ, and that the point stabiliser is primitive in its
natural action on n points. We prove that the maximum size of an irredundant
base of G is O(nβ), and in most cases O((logn)2). We also show that these bounds are best possible