Schur's colouring theorem for noncommuting pairs - Sanders, T (Cambridge University Press, 2019)
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 , yx. We show that c(G) → 0 if and only if k(G) → ∞.