  Free volume theory (FVT) is a versatile and tractable framework to predict
the phase behaviour of mixtures of platelets and non-adsorbing polymer chains
in a common solvent. Within FVT, three principal reference phases for the hard
platelets are considered: isotropic (I), nematic (N), and columnar (C). We
derive analytical expressions that enable us to systematically trace the
different types of phase coexistences revealed upon adding depletants and
confirm the predictive power of FVT by testing the calculated diagrams against
phase stability scenarios from computer simulation. A wide range of
\textit{multi}-phase equilibria is revealed, involving two-phase isostructural
transitions of all phase symmetries (INC) considered as well as the possible
three-phase coexistences. Moreover, we identify the system parameters, relative
disk shapes and colloid-polymer size ratios, at which four-phase equilibria are
expected. These involve a remarkable coexistence of all three phase states
commonly encountered in discotics including isostructural coexistences
I$_1$-I$_2$-N-C, I-N$_1$-N$_2$-C, and I-N-C$_1$-C$_2$.
