  The classical Ising model on the triangular lattice (we will call it I-3
model below), while simple in the nearest-neighbors (NN) only approximation,
becomes increasingly richer and more complex when further interactions are
incorporated. However, the studies so far have not been exhaustive, nor have
any attempts been made to estimate how realistic are the parameter ranges that
generate strong metamagnetism with a large number of magnetization steps. In
this study, we identify one such candidate, ErGa$_{2}$, a material known to
have one strong magnetization step, albeit some narrow steps below and above
cannot be confidently excluded. It has been established, and we can confirm the
same computationally, to have an easy axis perpendicular to the triangular Er
plane, with a strong anisotropy and with a large magnetic moment of
$9.5\:\mu_{B}$, making it a perfect implementation of the classical I-3 model.
In the first part of the analysis, we present the I-3 model with up to the
third nearest-neighbors in a range of parameters $J_{2}$ and $J_{3}$ ($J_{1}$
in this part is set to 1), and in some cases adding a rather small $J_{4}$ in
order to reveal new phases otherwise degenerate with some others. The richest
phase diagram is, not surprisingly, observed when all interactions are
antiferromagnetic (AF). Subsequently, a more realistic case, inspired by RKKY
and by our calculations for ErGa$_{2}$, where $J_{1},J_{2}>0$
(antiferromagnetic) and $J_{3},J_{4}<0$ (ferromagnetic), is presented. Finally,
we report our first-principles calculations of $J_{1-4}$ in ErGa$_{2}$ and
compared the phase diagram in the regime corresponding to the calculated values
with the experiment.
