We study lattice gas systems on the honeycomb lattice where particles exclude neighboring sites up to order $k$ ($k=1ldots5$) 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.
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