Irredundant bases for soluble groups

Let $\mathrm{\Delta }$ be a finite set and $G$ be a subgroup of $\mathrm{Sym}(\mathrm{\Delta })$. An irredundant base for $G$ is a sequence of points of $\mathrm{\Delta }$ yielding a strictly descending chain of pointwise stabilisers, terminating with the trivial group. Suppose that $G$ is primitive and soluble. We determine asymptotically tight bounds for the maximum length of an irredundant base for $G$. Moreover, we disprove a conjecture of Seress on the maximum length of an irredundant base constructed by the natural greedy algorithm, and prove Cameron’s Greedy Conjecture for $\mathrm{\mid }G\mathrm{\mid }$ odd.