  In the paper, we searh for monochromatic infinite additive structures
involving polynomials on $\mathbb{N}$. Ultimately, we can prove that for any
$r\in \mathbb{N}$, any distinct natural numbers $a,b$ and any $2$-coloring of
$\mathbb{N}$, there exist subsets $B,C\subset \mathbb{N}$ with $|B|=r$ and
$|C|=\infty$ such that there exists a color containing $B+aC$ and $B+bC$. In
fact, for the specific question considered by us, we give a complete answer.
