Monochromatic polynomial sumset structures on $\mathbb{N}$

In the paper, we search for monochromatic infinite additive structures involving polynomials over $\mathbb{N}$. It is proved that for any $r\in \mathbb{N}$, any two distinct natural numbers $a,b$, and any $2$-coloring of $\mathbb{N}$, there exist two sets $B,C\subset \mathbb{N}$ with $|B|=r$ and $|C|=\infty$ such that there exists some color containing $B+aC$ and $B+bC$.

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