Schur's colouring theorem for noncommuting pairs

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 yx. We show that c(G) -> 0 if and only if k(G) -> infinity.