We say that a first order formula Φ distinguishes a structure M over a 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+4 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 of two elements, 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+k 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.
"Descriptive complexity of finite structures: Saving the quantifier rank." J. Symbolic Logic 70 (2) 419 - 450, June 2005. https://doi.org/10.2178/jsl/1120224721