Monochromatic Polynomial sumset structures on N

In the paper, we searh for monochromatic infinite additive structures involving polynomials on N. Ultimately, we can prove that for any r∈ N, any distinct natural numbers $a,b$ and any $2$-coloring of N, there exist subsets B,C⊂ N with $|B|=r$ and |C|=∞ 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.