  We study RKKY interactions between local magnetic moments for both doped and
undoped graphene. We find in both cases that the interactions are primarily
ferromagnetic for moments on the same sublattice, and antiferromagnetic for
moments on opposite sublattices. This suggests that at sufficiently low
temperatures dilute magnetic moments embedded in graphene can order into a
state analogous to that of a dilute antiferromagnet. We find that in the
undoped case one expects no net magnetic moment, and demonstrate numerically
that this effect generalizes to ribbons where the magnetic response is
strongest at the edge, suggesting the possibility of an unusual spin-transfer
device. For doped graphene we find that moments at definite lattice sites
interact over longer distances than those placed in interstitial sites of the
lattice ($1/R^2$ vs. $1/R^3$) because the former support a Kohn anomaly that is
suppressed in the latter due to the absence of backscattering.
