  The purpose of this article is to show uniqueness theorems for meromorphic
mappings of C^m to CP^n with few hyperplanes H_j, j=1,...,q. It is well known
that uniqueness theorems hold for q \geq 3n+2.
  In this paper we show that for every nonnegative integer c there exists a
positive integer N(c), depending only on c in an explicit way, such that
uniqueness theorems hold if q\geq (3n+2 -c) and n\geq N(c). Furthermore, we
also show that the coefficient of n in the formula of q can be replaced by a
number which is strictly smaller than 3 for all n>>0. At the same time, a big
number of recent uniqueness theorems are generalized considerably.
