Element-Distinct Solution For Rado's Theorem

Abstract

In this paper, we present a simplified proof of Rado's Theorem and demonstrate that when an integer matrix $M$ satisfies the column condition and $M\mathbf x=\mathbf 0$ has an element-distinct solution on $\mathbb N$, then under any finite coloring of $\mathbb N$, the equation $M\mathbf x=\mathbf 0$ has a monochromatic element-distinct solution. This gives a positive answer to a problem of Di Nasso in 2016.