Descriptive complexity of finite structures: Saving the quantifier rank

We say that a first order formula Φ distinguishes a structure M over vocabulary L from another structure M ′ over the same vocabulary if Φ is true on M but false on M ′. A formula Φ defines an L-structure M if Φ distinguishes M from any other non-isomorphic L-structure M ′. A formula Φ identifies an n-element L-structure M if Φ distinguishes M from any other non-isomorphic n-element L-structure M ′. We prove that every n-element structure M is identifiable by a formula with quantifier rank less than (1− 1 2k )n+k2−k+2 and at most one quantifier alternation, where k is the maximum relation arity of M . Moreover, if the automorphism group of M contains no transposition, the same result holds for definability rather than identification. The Bernays-Schönfinkel class consists of prenex formulas in which the existential quantifiers all precede the universal quantifiers. We prove that every n-element structure M is identifiable by a formula in the Bernays-Schönfinkel class with less than (1− 1 2k2+2)n+ k quantifiers. If in this class of identifying formulas we restrict the number of universal quantifiers to k, then less than n − √n + k2 quantifiers suffice to identify M and, as long as we keep the number of universal quantifiers bounded by a constant, at total n − O(√n) quantifiers are necessary.