Equations resolving a conjecture of Rado on partition regularity

TL;DR

This paper proves Rado's conjecture that for each positive integer k, there exists a linear equation that is (k-1)-regular but not k-regular, advancing understanding of partition regularity in number theory.

Contribution

The paper confirms Rado's conjecture by explicitly constructing equations with the specified regularity properties, filling a long-standing gap in combinatorial number theory.

Findings

Constructed explicit equations with (k-1)-regular but not k-regular property.

Confirmed Rado's conjecture for all positive integers k.

Connected the result to classical problems in number theory.

Abstract

A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is (k-1)-regular but not k-regular. We prove this conjecture by showing that the equation $\sum_{i=1}^{k-1} \frac{2^i}{2^i-1} x_i = (-1 + \sum_{i=1}^{k-1} \frac{2^i}{2^i-1}) x_0$ has this property. This conjecture is part of problem E14 in Richard K. Guy's book "Unsolved problems in number theory", where it is attributed to Rado's 1933 thesis, "Studien zur Kombinatorik".