  In this paper we address the issue of the Gribov copies in SU(N), N>2,
Euclidean Yang-Mills theories quantized in the maximal Abelian gauge. A few
properties of the Gribov region in this gauge are established. Similarly to the
case of SU(2), the Gribov region turns out to be convex, bounded along the
off-diagonals directions in field space, and unbounded along the diagonal ones.
The implementation of the restriction to the Gribov region in the functional
integral is discussed through the introduction of the horizon function, whose
construction will be outlined in detail. The influence of this restriction on
the behavior of the gluon and ghost propagators of the theory is also
investigated together with a set of dimension two condensates.
