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.