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

2025-10-28

AI Summary

This paper establishes a structural equivalence between multiplayer parallel repetition games and high-dimensional extremal combinatorics. Methodologically, it rigorously models k-player, large-alphabet multi-prover games as forbidden subgraph problems on high-dimensional hypergraphs—marking the first precise translation of parallel repetition theory into extremal combinatorics. It extends the forbidden subgraph technique beyond the traditional two-player setting to arbitrary k ≥ 2 and large answer domains, integrating multi-prover interactive proof systems with high-dimensional combinatorial analysis. The main contributions are: (1) a bidirectional equivalence framework linking multiplayer parallel repetition and high-dimensional extremal combinatorics; (2) novel combinatorial tools for PCP theorems and hardness-of-approximation results; and (3) a unification of game theory, extremal combinatorics, and computational complexity, thereby broadening the applicability of classical techniques.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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 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.

Problem

• Establishes equivalence between multiplayer game parallel repetition
• Extends forbidden-subgraph technique to all k-player games
• Connects extremal combinatorics problems with multiplayer game theory

Innovation

• Equates multiplayer parallel repetition with extremal combinatorics
• Extends forbidden-subgraph technique to all k-player games
• Establishes new combinatorial connections for future progress