  We investigate higher-dimensional $\Delta$-systems indexed by finite sets of
ordinals, isolating a particular definition thereof and proving a
higher-dimensional version of the classical $\Delta$-system lemma. We focus in
particular on systems that consist of sets of ordinals, in which case useful
order-theoretic uniformities can be ensured. We then present three applications
of these higher-dimensional $\Delta$-systems to problems involving the
interplay between forcing and partition relations on the reals.
