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SCHUR’S COLOURING THEOREM FOR NONCOMMUTING PAIRS

Part of: Graph theory

Published online by Cambridge University Press:  11 April 2019

TOM SANDERS Open the ORCID record for TOM SANDERS [Opens in a new window]
TOM SANDERS*
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK email [email protected]
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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 $xy\neq yx$. We show that $c(G)\rightarrow 0$ if and only if $k(G)\rightarrow \infty$.

Keywords

Schur’s colouring theoremcommuting probabilitynon-Abelian group

MSC classification

Secondary: 05C15: Coloring of graphs and hypergraphs
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 3 , December 2019 , pp. 446 - 452
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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