Model theoretic characterizations of large cardinals

We consider compactness characterizations of large cardinals. Based on results of Benda [Ben78], we study compactness for omitting types in various logics. In L-kappa,L-kappa, this allows us to characterize any large cardinal defined in terms of normal ultrafilters, and we also analyze second-order and sort logic. In particular, we give a compactness for omitting types characterization of huge cardinals, which have consistency strength beyond Vopenka's Principle.