  We show that a finite coloring of an amenable group contains `many'
monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with
more variables. This gives the first combinatorial proof and extensions of
Bergelson and McCutcheon's non-commutative Schur theorem. Our main new tool is
the introduction of what we call `quasirandom colorings,' a condition that is
automatically satisfied by colorings of quasirandom groups, and a reduction to
this case.
