  Using Talagrand's concentration inequality on the discrete cube {0,1}^m we
show that given a real-valued function Z(x)on {0,1}^m that satisfies certain
monotonicity conditions one can control the deviations of Z(x) above its median
by a local Lipschitz norm of Z(x) at the point x. As one application, we give a
simple proof of a nearly optimal deviation inequality for the number of
k-cycles in a random graph.
