  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.
