On the minimal degree and base size of finite primitive groups

Let [Formula: see text] be a finite permutation group acting on [Formula: see text]. A base for [Formula: see text] is a subset [Formula: see text] such that the pointwise stabilizer [Formula: see text] is the identity. The base size of [Formula: see text], denoted by [Formula: see text], is the cardinality of the smallest possible base. The minimal degree of [Formula: see text], denoted by [Formula: see text], is the smallest cardinality of the support of a non-trivial element of [Formula: see text]. In this paper, we establish a new upper bound for [Formula: see text] when [Formula: see text] is primitive, and subsequently prove that if [Formula: see text] is a primitive group different from the Mathieu group of degree [Formula: see text], then [Formula: see text], where [Formula: see text] is the degree of [Formula: see text]. This bound is best possible, up to a multiplicative constant.