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

TL;DR

This paper establishes a deep equivalence between high-dimensional extremal combinatorics problems and the parallel repetition of multiplayer games, extending previous techniques and creating new links between these fields.

Contribution

It generalizes the forbidden-subgraph technique to all k-player games and connects parallel repetition to high-dimensional combinatorial problems, opening new research avenues.

Findings

Extended the forbidden-subgraph technique to all k-player games.

Established equivalences between combinatorics problems and multiplayer game repetition.

Created new connections that may facilitate future advances in both fields.

Abstract

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 H\k{a}z{\l}a , 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.