  Correlated systems with hexagonal layered structures have come to fore with
renewed interest in Cobaltates, transition-metal dichalcogenides and GdI2.
While superconductivity, unusual metal and possible exotic states (prevented
from long range order by strong local fluctuations) appear to come from
frustration and correlation working in tandem in such systems, they freeze at
lower temperature to crystalline states. The underlying effective Hamiltonian
in some of these systems is believed to be the Falicov-Kimball model and
therefore, a thorough study of the ground state of this model and its extended
version on a non-bipartite lattice is important. Using a Monte Carlo search
algorithm, we identify a large number of different possible ground states with
charge order as well as valence and metal-insulator transitions. Such competing
states, close in energy, give rise to the complex charge order and other broken
symmetry structures as well as phase segregations observed in the ground state
of these systems.
