Greedy bases and relational complexity of diagonal type groups

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.