Solution Regularity of k-partite Linear Systems -- Variant of Rado's Theorem

Hongyi Zhou

Published 2021-07-04Version 1

Ramsey theory is a modern and arresting field of mathematics with a century of history. The basic paradigm of Ramsey Theory is that any sufficiently large regular structure contains some highly ordered substructure. One number theoretic Ramseyan result is the Rado's Theorem. In this research, I extended the Rado's Theorem to a more general model. Given $k \ge 2$, there are some necessary and sufficient conditions such that for a system of linear equations to be semi-regular on $\mathbb{Z}$. In this article, I will show the proof to this extended theorem with a combination of conventional strategies (applied to prove Rado's Theorem) and new perspectives (in linear algebra and number theory). Some polynomials methods are applied here to make use of the existence of infinitely many desired prime numbers.