Element-Distinct Solution For Rado's Theorem

Ningyuan Yang; Tianyi Tao

Element-Distinct Solution For Rado's Theorem

Combinatorics 2024-12-02 v2

Authors: Ningyuan Yang , Tianyi Tao

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.

Keywords

ramsey theory and colorings

Cite

@article{arxiv.2411.17456,
  title  = {Element-Distinct Solution For Rado's Theorem},
  author = {Ningyuan Yang and Tianyi Tao},
  journal= {arXiv preprint arXiv:2411.17456},
  year   = {2024}
}

Comments

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 I believe the paper should be withdrawn

Element-Distinct Solution For Rado's Theorem

Abstract

Keywords

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