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On Cameron's Greedy Conjecture

Coen del Valle, Colva M. Roney-Dougal

TL;DR

The paper advances Cameron's Greedy Conjecture for two natural primitive actions of symmetric groups: on $r$-subsets and on partitions. It introduces a meta-greedy analysis for the $r$-subset action to prove $\mathcal{G}(G)\le \tfrac{17}{10}\,b(G)$ when $n\ge 4r^2$, and develops a 3D intersection-array framework to bound greedy bases for the partition action, obtaining $\mathcal{G}(G)\le 11\,b(G)$. Together these results show the conjecture holds for these broad families up to a finite set of exceptions, and provide explicit constants that quantify the proximity of greedy bases to minimal bases in primitive groups. The work combines orbit-structure analysis, generalized intersection-array theory, and targeted computations to derive practical, provable bounds on greedy bases.

Abstract

A base for a permutation group $G$ acting on a set $Ω$ is a subset $\mathcal{B}$ of $Ω$ whose pointwise stabiliser $G_{(\mathcal{B})}$ is trivial. There is a natural greedy algorithm for constructing a base of relatively small size. We write $\mathcal{G}(G)$ the maximum size of a base it produces, and $b(G)$ for the size of the smallest base for $G$. In 1999, Peter Cameron conjectured that there exists an absolute constant $c$ such that every finite primitive group $G$ satisfies $\mathcal{G}(G)\leq cb(G)$. We show that if $G$ is $\mathrm{S}_n$ or $\mathrm{A}_n$ acting primitively then either Cameron's Greedy Conjecture holds for $G$, or $G$ falls into one class of possible exceptions.

On Cameron's Greedy Conjecture

TL;DR

The paper advances Cameron's Greedy Conjecture for two natural primitive actions of symmetric groups: on -subsets and on partitions. It introduces a meta-greedy analysis for the -subset action to prove when , and develops a 3D intersection-array framework to bound greedy bases for the partition action, obtaining . Together these results show the conjecture holds for these broad families up to a finite set of exceptions, and provide explicit constants that quantify the proximity of greedy bases to minimal bases in primitive groups. The work combines orbit-structure analysis, generalized intersection-array theory, and targeted computations to derive practical, provable bounds on greedy bases.

Abstract

A base for a permutation group acting on a set is a subset of whose pointwise stabiliser is trivial. There is a natural greedy algorithm for constructing a base of relatively small size. We write the maximum size of a base it produces, and for the size of the smallest base for . In 1999, Peter Cameron conjectured that there exists an absolute constant such that every finite primitive group satisfies . We show that if is or acting primitively then either Cameron's Greedy Conjecture holds for , or falls into one class of possible exceptions.

Paper Structure

This paper contains 5 sections, 28 theorems, 100 equations, 1 table.

Table of Contents

  1. Introduction
  2. $\mathrm{S}_n$ and $\mathrm{A}_n$ on $r$-subsets
  3. $\mathrm{S}_n$ and $\mathrm{A}_n$ on partitions
  4. Intersection arrays and their properties
  5. Proof of Theorem \ref{['camskl']}

Key Result

Theorem 1.2

Let $\mathcal{F}$ be the family of almost simple primitive groups with alternating socle, excluding those acting on ${[n]\choose r}$ with $4r^2>n>2r$. Then $\mathcal{F}$ is an $11$-ravenous family.

Theorems & Definitions (54)

  • Conjecture 1.1: Cameron's Greedy Conjecture cam
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 44 more