# A Combinatorial Problem from Group Theory

## TL;DR

This paper proves Keller's conjecture for the general case of an n-by-infinite matrix related to group actions and introduces a combinatorial game that further generalizes the problem.

## Contribution

It extends Keller's conjecture proof from the case n=4 to all n and proposes a new combinatorial game that generalizes the original problem.

## Key findings

- Keller's conjecture is true for all n
- A combinatorial game version of the conjecture is introduced
- The proof involves group action orbit analysis

## Abstract

Keller proposed a combinatorial conjecture on construction of an n-by-infinite matrix, which comes from showing the existence of many orbits of different sizes in certain linear group actions. He proved it for the case n=4, and we show that conjecture is true in the general case. We also propose a combinatorial game version of the conjecture which even further generalizes the problem.