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 $M\mathbf{x}=0$ has an element-distinct solution on $\mathbb{N}$, then under any finite coloring of $\mathbb{N}$, the equation $M\mathbf{x}=0$ has a monochromatic element-distinct solution. This gives a positive answer to a problem of Di Nasso in 2016.

The main conclusions presented in this paper have been proven previously by others, and the original author has contacted me. I believe the paper should be withdrawn