  We report the grand-canonical solution of a ternary mixture of discrete hard
spheres defined on a Husimi lattice built with cubes, which provides a
mean-field approximation for this system on the cubic lattice. The mixture is
composed by point-like particles (0NN) and particles which exclude up to their
first (1NN) and second neighbors (2NN), with activities $z_0$, $z_1$ and $z_2$,
respectively. Our solution reveals a very rich thermodynamic behavior, with two
solid phases associated with the ordering of 1NN ($S1$) or 2NN particles
($S2$), and two fluid phases, being one regular ($RF$) and the other
characterized by a dominance of 0NN particles ($F0$ phase). However, in most
part of the phase diagram these fluid ($F$) phases are indistinguishable.
Discontinuous transitions are observed between all the four phases, yielding
several coexistence surfaces in the system, among which a fluid-fluid and a
solid-solid demixing surface. The former one is limited by a line of critical
points and a line of triple points (where the phases $RF$-$F0$-$S2$ coexist),
both meeting at a special point, after which the fluid-fluid coexistence
becomes metastable. Another line of triple points is found, connecting the
$F$-$S1$, $F$-$S2$ and $S1$-$S2$ coexistence surfaces. A critical $F$-$S1$
surface is also observed meeting the $F$-$S1$ coexistence one at a line of
tricritical points. Furthermore, a thermodynamic anomaly characterized by
minima in isobaric curves of the total density of particles is found, yielding
three surfaces of minimal density in the activity space, depending on which
activity is kept fixed during its calculation.
