The \(2\)-color Rado Number of \(x_1 +x_2 + \dots + x_n = y_1 +y_2+ \ldots + y_k\)

Abstract

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