  Given a relational structure M on n elements, let D(M) be the minimum
quantifier rank of a first order formula identifying M up to isomorphism in the
class of n-element structures. The obvious upper bound is D(M)\le n. We show
that if the relations in M have arity at most k, then
D(M)<(1-\frac{1}{2k})n+k^2-k+2. The coefficient at n, which equals
1-\frac{1}{2k}, is probably not best possible but this is the first known bound
having it strictly below 1 (for fixed k).
  If one is content to have the worse coefficient 1-\frac{1}{2k^2+2}, then one
can choose an identifying formula of a very special form: a prenex formula with
at most one quantifier alternation.
  A few other results in this vein are presented.
