Higher-dimensional Delta-systems

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.