  Uniform bounds are obtained using the auxiliary Monge-Amp\`ere equation
method for solutions of very general classes of fully non-linear partial
differential equations, assuming the existence of a ${C}$-subsolution in the
sense of G. Sz\'ekelyhidi and B. Guan. The main advantage over previous
estimates is in the K\"ahler case, where the new estimates remain uniform as
the background metric is allowed to degenerate.
