The $2$-color Rado Number of $x_1+x_2+\cdots +x_n=y_1+y_2+\dots +y_k$

Abstract

In $1982$, Beutelspacher and Brestovansky determined the $2$-color Rado number of the equation $$x_1+x_{2}x+\dots +x_{m-1}=x_m$$ for all $m\ge 3$. Here we extend their result by determining the 2-color Rado number of the equation $$x_1+x_2+\cdots +x_n=y_1+y_2+\dots +y_k$$ for all $n\ge 2$ and $k\ge 2$. As a consequence, we determine the 2-color Rado number of $$x_1+x_2+\dots +x_n=a_1{y}_1+\cdots +a_{\ell }{y}_{\ell }$$ in all cases where $n\ge 2$ and $n\ge a_1+\cdots +a_{\ell }$, and in most cases where $n\ge 2$ and $2n\ge a_1+\cdots +a_{\ell }$.