Multiplicity of monochromatic solutions to x+yz

For integers n>=1 and k>=0, let M"k(n) represent the minimum number of monochromatic solutions to x+y=0, M"k(n)=Cn^3(1+o"k(1)), where C=112(1+22)^2~0.005686. A structural result is also proven, which can be used to determine the exact value of M"k(n) for given k and n.