Greedy base sizes for sporadic simple groups

TL;DR

This paper proves that for all almost simple primitive groups with sporadic simple socles, the greedy algorithm produces a base size equal to the minimal possible, confirming a long-standing conjecture.

Contribution

It establishes that the greedy base construction yields minimal bases for all such groups, resolving a key conjecture in permutation group theory.

Findings

The greedy algorithm's base size matches the minimal base size for these groups.

Confirms the conjecture that $\, ext{maximal greedy base size} = ext{minimal base size}$.

Provides a complete classification for sporadic simple groups.

Abstract

A base for a permutation group $G$ acting on a set $\Omega$ is a sequence $\mathcal{B}$ of points of $\Omega$ such that the pointwise stabiliser $G_{\mathcal{B}}$ is trivial. Denote the minimum size of a base for $G$ by $b(G)$. There is a natural greedy algorithm for constructing a base of relatively small size; denote by $\mathcal{G}(G)$ the maximum size of a base it produces. Motivated by a long-standing conjecture of Cameron, we determine $\mathcal{G}(G)$ for every almost simple primitive group $G$ with socle a sporadic simple group, showing that $\mathcal{G}(G)=b(G)$.