Monochromatic Sums and Products of Polynomials

Abstract: We show that the pattern ${x,x+y,xy}$ is partition regular over the space of formal integer polynomials of degree at least one with zero constant term, with primitive recursive bounds. This provides a new proof for the partition regularity of ${x,x+y,xy}$ over $\mathbb{N}$, which gives the first primitive recursive bound.