On relational complexity and base size of finite primitive groups

TL;DR

This paper establishes a logarithmic upper bound on the size of irredundant bases for finite primitive groups that are not large base, with implications for relational complexity and computational efficiency.

Contribution

It provides the first logarithmic bound on irredundant base size for such groups, improving understanding of their structure and computational properties.

Findings

Irredundant base size is at most 5 log n for these groups.

Relational complexity is bounded by 5 log n + 1.

A base of size at most 5 log n can be computed in polynomial time.

Abstract

In this paper we show that if $G$ is a primitive subgroup of $S_{n}$ that is not large base, then any irredundant base for $G$ has size at most $5 \log n$. This is the first logarithmic bound on the size of an irredundant base for such groups, and is best possible up to a small constant. As a corollary, the relational complexity of $G$ is at most $5 \log n+1$, and the maximal size of a minimal base and the height are both at most $5 \log n.$ Furthermore, we deduce that a base for $G$ of size at most $5 \log n$ can be computed in polynomial time.