Schur’s Colouring Theorem for Noncommuting pairs

Abstract

For G a finite non-Abelian group we write cpGq for the probability that two randomly chosen elements commute and kpGq for the largest integer such that any kpGqcolouring of G is guaranteed to contain a monochromatic quadruple px, y, xy, yxq with xy ‰ yx. We show that cpGq Ñ 0 if and only if kpGq Ñ 8.