Element-Distinct Solution For Rado's Theorem

TL;DR

This paper provides a simplified proof of Rado's Theorem, showing that certain linear equations with element-distinct solutions guarantee monochromatic solutions under any finite coloring, solving a 2016 problem.

Contribution

It introduces a simplified proof of Rado's Theorem and establishes the existence of monochromatic element-distinct solutions for specific matrices, addressing a problem posed in 2016.

Findings

Simplified proof of Rado's Theorem

Monochromatic element-distinct solutions exist under finite colorings

Addresses a problem posed by Di Nasso in 2016

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.