The pigeonhole principle and multicolor Ramsey numbers

A standard proof of Schur's Theorem yields that any $r$-coloring $\{1,2,\dots,R_r-1\}$ a monochromatic solution to $x+y=z$, where $R_r$ is the classical $r$-color Ramsey number, minimum $N$ such complete graph on vertices triangle. We explore generalizations and modifications this result in higher dimensional integer lattices, showing particular if $k\geq d+1$, then $\{1,2,\dots,R_r(k)^d-1\}^d$ $x_1+\cdots+x_{k-1}=x_k$ with $\{x_1,\dots,x_d\}$ linearly independent, $R_r(k)$ analogous number which triangles are replaced by graphs $k$ vertices. also obtain computational results examples case $d=2$, $k=3$, $r\in\{2,3,4\}$.