  We first present a simple proof that for any bipartite lattice at half
filling the RKKY interaction is antiferromagnetic between impurities on
opposite (i.e., A and B) sublattices and is ferromagnetic between impurities on
the same sublattices. This result is valid on all length scales. We then focus
on the honeycomb lattice and examine the theorem in the long distance limit by
performing the low energy calculation using Dirac electrons. To find the
universal (cutoff free) result we perform the calculation in smooth cutoff
schemes, as we show that the calculation based on a sharp cutoff leads to wrong
results. We also find the long distance behavior of the RKKY interaction
between "plaquette" impurities in both coherent and incoherent regimes.
