  We consider the problem of testing whether a correlation matrix of a
multivariate normal population is the identity matrix. We focus on sparse
classes of alternatives where only a few entries are nonzero and, in fact,
positive. We derive a general lower bound applicable to various classes and
study the performance of some near-optimal tests. We pay special attention to
computational feasibility and construct near-optimal tests that can be computed
efficiently. Finally, we apply our results to prove new lower bounds for the
clique number of high-dimensional random geometric graphs.
