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

TL;DR

This paper extends Rado's theorem to k-partite systems, establishing conditions for semi-monochromatic solutions in colored sets, and generalizes to algebraic number fields using polynomial root methods.

Contribution

It introduces a variant of Rado's theorem for k-partite systems, linking semi-monochromatic solutions to roots of linear polynomials and extending to algebraic number fields.

Findings

Conditions for semi-monochromatic solutions in k-partite systems

Connection between solutions and roots of linear polynomials

Generalization to algebraic number fields

Abstract

A classical question in combinatorial number theory asks whether an equation has a solution inside a particular subset of its domain. The Rado's Theorem gives a set of necessary and sufficient conditions for a systems of linear equations to have a monochromatic solution whenever the positive integers are finitely colored. In this paper, we provide a variant of this theorem. For $k \ge 2$, we present conditions such that, when the set of variables is partitioned into $k$ subsets, there is a solution such that the variables of each subset are monochromatic, which we call a semi-monochromatic solution. We adopt the smod $p$ coloring by Graham, Rothschild, and Spencer but turn the existence of semi-monochromatic solution into the existence of common roots of linear polynomials. With this idea, one can further generalize the theorem to systems of linear equations over general algebraic number fields.