Cameron’s greedy conjecture for finite primitive permutation groups

Establish the existence of an absolute constant c such that for every finite primitive permutation group G, the greedy base size G(G) is at most c times the base size b(G). Here b(G) denotes the minimum size of a base (a sequence of points whose pointwise stabilizer in G is trivial), and G(G) denotes the maximum size of a base produced by the greedy algorithm that iteratively selects a point from a largest G-orbit and then, at each subsequent step, selects a point in a largest orbit of the stabilizer of the previously chosen points until the pointwise stabilizer is trivial.

Background

The paper studies bases in permutation groups and a natural greedy algorithm introduced by Blaha for constructing small bases. The base size b(G) is the minimum number of points needed so that the pointwise stabilizer in G is trivial, while the greedy base size G(G) is the maximum size of a base produced by the greedy algorithm that repeatedly chooses points from largest orbits.

Cameron’s greedy conjecture posits a uniform linear bound between G(G) and b(G) for all finite primitive permutation groups. The authors focus on determining G(G) for almost simple primitive groups with sporadic socle, showing that G(G)=b(G) in these cases, and note that resolving Cameron’s conjecture remains open more generally.