Greedy base sizes for sporadic simple groups

Abstract A base for a permutation group 𝐺 acting on a set Ω is a sequence ℬ of points of Ω such that the pointwise stabiliser <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mi mathvariant="script">B</m:mi> </m:msub> </m:math> G_{\mathcal{B}} is trivial. Denote the minimum size of a base for 𝐺 by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> b(G) . There is a natural greedy algorithm for constructing a base of relatively small size; denote by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">G</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{G}(G) the maximum size of a base it produces. Motivated by a long-standing conjecture of Cameron, we determine <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">G</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{G}(G) for every almost simple primitive group 𝐺 with socle a sporadic simple group, showing that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="script">G</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \mathcal{G}(G)=b(G) .