Some Extensions of Finite Sum Theorem

Abstract: The paper gives some multi-dimensional extensions of Hindman's finite sum theorem. In particular, by the method of this paper, we prove that for any finite coloring of $\mathbb N$, there are $a,b\in \mathbb N$ such that there exist (infinitely many) pairs $(x,y),(u,v)\in \mathbb N^2$ such that the two sets ${ax,ay,xy,a(x+y)}$ and ${u+b,v+b,uv+b,u+v}$ are monochromatic.