Greedy bases and relational complexity of diagonal type groups

TL;DR

This paper proves Cameron's conjecture for primitive diagonal type groups by analyzing the greedy base algorithm and explores the unbounded relational complexity of these groups.

Contribution

It determines the size of bases returned by the greedy algorithm for primitive diagonal type groups and establishes that their relational complexity is unbounded.

Findings

The size of the greedy algorithm's bases for these groups is explicitly characterized.

Cameron's conjecture is confirmed for primitive diagonal type groups.

Relational complexity of these groups is at least 4 and unbounded.

Abstract

A base for a subgroup $G$ of $\mathrm{Sym}(\Omega)$ is a sequence of elements of $\Omega$ with trivial pointwise stabiliser. The size of the smallest base for $G$ is denoted $b(G)$. There is a natural greedy algorithm to compute a base for $G$, and it was conjectured by Cameron in 1999 that there exists an absolute constant $c$ such that if $G$ is primitive then any base returned by this algorithm has size at most $cb(G)$. In this paper we determine the size of every base returned by the greedy algorithm when $G$ is a primitive group of diagonal type, and hence prove Cameron's conjecture for these groups. The relational complexity $\mathrm{RC}(G)$ of $G$ is a measure of the way in which the orbits of $G$ on $\Omega^k$ for various $k$ determine the action of $G$ on $\Omega$. Very few precise values of relational complexity are known, and in particular it is not known which primitive groups have relational complexity $3$. In this paper we prove that if $G$ is primitive of diagonal type then $\mathrm{RC}(G) \geqslant 4$, that this lower bound is attained by infinitely many such $G$, and that the relational complexity of the groups of diagonal type is unbounded.