Schur's colouring theorem for non-commuting 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 to yx. We show that c(G) tends to 0 if and only
if k(G) tends to infinity.