Monochromatic non-commuting products

Abstract: 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 callquasirandom colorings,' a condition that is automatically satisfied by colorings of quasirandom groups, and a reduction to this case.