Multiplayer Parallel Repetition Is the Same as High-Dimensional Extremal Combinatorics

We show equivalences between several high-dimensional problems in extremal combinatorics and parallel repetition of multiplayer (multiprover) games over large answer alphabets. This extends the forbidden-subgraph technique, previously studied by Verbitsky (Theoretical Computer Science 1996), Feige and Verbitsy (Combinatorica 2002), and Hązła , Holenstein and Rao (2016), to all $k$-player games, and establishes new connections to problems in combinatorics. We believe that these connections may help future progress in both fields.