In recent joint work with Asger Toernquist, we showed how to construct
definable maximal discrete sets in forcing extensions of L,
in particular in the Sacks and Miller extension. In particular, the
existence of such sets is consistent with V \neq L.
In this talk I shall show the stronger result that the existence of
definable discrete sets is consistent with large continuum. In the
process, I show an interesting generalization of Galvin's theorem. In
particular, this applies
to the example of maximal orthogonal families of measures (mofs).
One might hope for a simpler way of constructing a mof in a model with
large continuum: to find an indestructible such family in L. While such
an approach is possible e.g. for maximal cofinitary groups, this is
impossible for mofs.