Monochromatic Polynomial sumset structures on $\mathbb{N}$: 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 $\mathbb{N}$, there exists two infinite subsets $B,C$ of $\mathbb{N}$ such that the set $\{P(b)+P(c):b\in B,c\in C\}$ 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 $f(\mathbb{N})\setminus \mathbb{N}$ is finite. We use ultrafilter techniques to prove our result.