Key points are not available for this paper at this time.
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.
Lian et al. (Mon,) studied this question.