  We study lattice gas systems on the honeycomb lattice where particles exclude
neighboring sites up to order $k$ ($k=1\ldots5$) from being occupied by another
particle. Monte Carlo simulations were used to obtain phase diagrams and
characterize phase transitions as the system orders at high packing fractions.
For systems with first neighbors exclusion (1NN), we confirm previous results
suggesting a continuous transition in the 2D-Ising universality class.
Exclusion up to second neighbors (2NN) lead the system to a two-step melting
process where, first, a high density columnar phase undergoes a first order
phase transition with non-standard scaling to a solid-like phase with short
range ordered domains and, then, to fluid-like configurations with no sign of a
second phase transition. 3NN exclusion, surprisingly, shows no phase transition
to an ordered phase as density is increased, staying disordered even to packing
fractions up to 0.98. The 4NN model undergoes a continuous phase transition
with critical exponents close to the 3-state Potts model. The 5NN system
undergoes two first order phase transitions, both with non-standard scaling.
We, also, propose a conjecture concerning the possibility of more than one
phase transition for systems with exclusion regions further than 5NN based on
geometrical aspects of symmetries.

Soft Condensed Matter;Statistical Mechanics;Computational Physics