SCHUR'S COLOURING THEOREM FOR NONCOMMUTING PAIRS

TOM SANDERS
2019 Bulletin of the Australian Mathematical Society  
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 $k(G)\rightarrow \infty$ .
doi:10.1017/s0004972719000406 fatcat:43p66nz44fcwvhj66tunlzwam4