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arXiv · 2605.16032

Greedy bases and relational complexity of diagonal type groups

Hong Yi Huang·Colva M. Roney-Dougal
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Abstract

A base for a subgroup $G$ of $\mathrm{Sym}(Ω)$ is a sequence of elements of $Ω$ 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 $Ω^k$ for various $k$ determine the action of $G$ on $Ω$. 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.

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BibTeXRIS

Hong Yi Huang, Colva M. Roney-Dougal. 2026-05-15. Greedy bases and relational complexity of diagonal type groups. https://arxiv.org/abs/2605.16032

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