Monochromatic Polynomial sumset structures on N: an ultrafilter proof

Recently, using machinery's from Ergodic theory, Z. Lian, and R. Xiao proved if P is any polynomial with no constant term, then for every finite coloring of N, there exists two infinite subsets B, C of N such that the set \P (b) +P (c): b B, c C\ is monochromatic. In this article we improve their result by proving that instead of taking such polynomials we can choose any function f having the property that f (N) N is finite. We use ultrafilter techniques to prove our result.