  We consider the high-energy limits of the colour ordered four-, five- and
six-gluon MHV amplitudes of the maximally supersymmetric QCD in the multi-Regge
kinematics where all the gluons are strongly ordered in rapidity. We show that
various building blocks occurring in the Regge factorisation (the Regge
trajectory, the coefficient functions and the Lipatov vertex) satisfy an
iterative structure very similar to the Bern-Dixon-Smirnov (BDS) ansatz. This
iterative structure, combined with the universality of the building blocks,
enables us to show that in the Euclidean region any two- and three-loop
amplitude in multi-Regge kinematics is guaranteed to satisfy the BDS ansatz. We
also consider slightly more general kinematics where the strong rapidity
ordering applies to all the gluons except the two with either the largest or
smallest rapidities, and we derive the iterative formula for the associated
coefficient function. We show that in this kinematic limit the BDS ansatz is
also satisfied. Finally, we argue that only for more general kinematics - e.g.
with three gluons having similar rapidities, or where the two central gluons
have similar rapidities - can a disagreement with the BDS ansatz arise.
