  We study the ground states of cold atoms in the tight-binding bands built
from p-orbitals on a two dimensional honeycomb optical lattice. The band
structure includes two completely flat bands. Exact many-body ground states
with on-site repulsion can be found at low particle densities, for both
fermions and bosons. We find crystalline order at n=1/6 with a $\sqrt{3} \times
\sqrt{3}$ structure breaking a number of discrete lattice symmetries. In
fermionic systems, if the repulsion is strong enough, we find the bonding
strength becomes \emph{dimerized} at n=1/2. Experimental signatures of
crystalline order can be detected through the noise correlations in time of
flight experiments.
