On relational complexity and base size of finite primitive groups

Abstract

In this paper we show that if G is a primitive subgroup of Sn 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 multiplicative 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.