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

Abstract

In 1982, Beutelspacher and Brestovansky determined the 2-color Rado number of the equation $$x_1+x_2+\cdots +x_{m-1}=x_m$$ for all $m\geq 3.$ Here we extend their result by determining the 2-color Rado number of the equation $$x_1+x_2+\cdots +x_n=y_1+y_2+\cdots +y_k$$ for all $n\geq 2$ and $k\geq 2.$ As a consequence, we determine the 2-color Rado number of $$x_1+x_2+\cdots +x_n=a_1y_1+\cdots +a_{\ell}y_{\ell}$$ in all cases where $n\geq 2$ and $n\geq a_1+\cdots +a_{\ell},$ and in most cases where $n\geq 2$ and $2n\geq a_1+\cdots +a_{\ell}.$