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

Kunal Mittal

TL;DR

This paper uncovers deep connections between parallel repetition in multiplayer games and high-dimensional extremal combinatorics by employing the forbidden-subgraph framework. It proves that, for a game G with val(G) < 1, the n-fold repetition value is controlled by extremal quantities E_Q(n), r_q(n), and r_grid(F,k,n), with val(G^n) ≤ E_Q(n) (uniform distributions) and tightness in the large-alphabet regime. It further links these bounds to central combinatorial problems: for combinatorial lines via density-Hales-Jewett bounds, for squares via the GHZ-related construction, and for grids via affine subspaces, thereby enabling a two-way transfer of techniques between interactive proofs/communication complexity and extremal/additive combinatorics. The work suggests that progress in one field can translate into meaningful advances in the other, enabling new avenues to attack longstanding questions in high-dimensional combinatorics using parallel-repetition tools and vice versa.

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.

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

TL;DR

This paper uncovers deep connections between parallel repetition in multiplayer games and high-dimensional extremal combinatorics by employing the forbidden-subgraph framework. It proves that, for a game G with val(G) < 1, the n-fold repetition value is controlled by extremal quantities E_Q(n), r_q(n), and r_grid(F,k,n), with val(G^n) ≤ E_Q(n) (uniform distributions) and tightness in the large-alphabet regime. It further links these bounds to central combinatorial problems: for combinatorial lines via density-Hales-Jewett bounds, for squares via the GHZ-related construction, and for grids via affine subspaces, thereby enabling a two-way transfer of techniques between interactive proofs/communication complexity and extremal/additive combinatorics. The work suggests that progress in one field can translate into meaningful advances in the other, enabling new avenues to attack longstanding questions in high-dimensional combinatorics using parallel-repetition tools and vice versa.

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 -player games, and establishes new connections to problems in combinatorics. We believe that these connections may help future progress in both fields.

Paper Structure

This paper contains 13 sections, 15 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Parallel Repetition of Multiplayer Games
  3. Arithmetic Combinatorics
  4. Arithmetic Progressions.
  5. Combinatorial Lines.
  6. Corners and Squares.
  7. This Work
  8. Multiplayer Games
  9. Forbidden Subgraphs
  10. Connections to Extremal Combinatorics
  11. Density Hales-Jewett Theorem
  12. Square-Free sets over F_2n
  13. Grid-Free sets over Fn

Key Result

Theorem 1.1

For every positive integers $r,k\in \mathbb{N}$, there exists an integer $n\in \mathbb{N}$, such that if the numbers $[n] = \left\{ 1,2,\dots,n \right\}$ are colored with $r$ colors, then at least one of the colors contains a $k$-term arithmetic progression. By a $k$-term arithmetic progression, we for integers $a,d$ with $d\not=0$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • ...and 29 more