Irredundant bases for soluble groups - Brenner, Sofia; del Valle, Coen; Roney‐Dougal, Colva M. (The Bulletin of the London Mathematical Society, 2025)
Let Δ$Δ$ be a finite set and G$G$ be a subgroup of Sym(Δ)$\operatorname{Sym}(Δ)$. An irredundant base for G$G$ is a sequence of points of Δ$Δ$ yielding a strictly descending chain of pointwise stabilisers, terminating with the trivial group. Suppose that G$G$ is primitive and soluble. We determine asymptotically tight bounds for the maximum length of an irredundant base for G$G$. 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 |G|$|G|$ odd.