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 (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.