  In the Horndeski's most general scalar-tensor theories with second-order
field equations, we derive the conditions for the avoidance of ghosts and
Laplacian instabilities associated with scalar, tensor, and vector
perturbations in the presence of two perfect fluids on the flat
Friedmann-Lemaitre-Robertson-Walker (FLRW) background. Our general results are
useful for the construction of theoretically consistent models of dark energy.
We apply our formulas to extended Galileon models in which a tracker solution
with an equation of state smaller than -1 is present. We clarify the allowed
parameter space in which the ghosts and Laplacian instabilities are absent and
we numerically confirm that such models are indeed cosmologically viable.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Phenomenology;High Energy Physics - Theory