Monochromatic non-commuting products

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.