# Schur's colouring theorem for non-commuting pairs

## TL;DR

This paper investigates the relationship between the probability of elements commuting in finite non-Abelian groups and the minimal coloring size needed to guarantee monochromatic quadruples with non-commuting pairs.

## Contribution

It establishes a precise equivalence between the decay of commutativity probability and the unboundedness of the coloring number for such groups.

## Key findings

- c(G) tends to 0 iff k(G) tends to infinity
- Provides a new link between algebraic structure and combinatorial coloring
- Characterizes the asymptotic behavior of non-Abelian groups

## Abstract

For G a finite non-Abelian group we write c(G) for the probability that two randomly chosen elements commute and k(G) for the largest integer such that any k(G)-colouring of G is guaranteed to contain a monochromatic quadruple (x,y,xy,yx) with xy not equal to yx. We show that c(G) tends to 0 if and only if k(G) tends to infinity.