Resolving a Conjecture on Degree of Regularity of Linear Homogeneous Equations

TL;DR

This paper proves that a specific family of linear equations has a degree of regularity of n-1, confirming a conjecture by Fox and Radoicic and extending previous results on Rado's conjecture.

Contribution

The paper establishes that Fox and Radoicic's family of equations have a degree of regularity of n-1, confirming their conjecture and extending the understanding of linear homogeneous equations.

Findings

Confirmed the degree of regularity as n-1 for Fox and Radoicic's equations

Extended previous results on Rado's conjecture

Provided new insights into the structure of regular linear equations

Abstract

A linear equation is $r$-regular, if, for every $r$-coloring of the positive integers, there exist positive integers of the same color which satisfy the equation. In 2005, Fox and Radoicic conjectured that the equation $x_1 + 2x_2 + \cdots + 2^{n-2}x_{n-1} - 2^{n-1}x_n = 0$, for any $n \geq 2$, has a degree of regularity of $n-1$, which would verify a conjecture of Rado from 1933. Rado's conjecture has since been verified with a different family of equations. In this paper, we show that Fox and Radoicic's family of equations indeed have a degree of regularity of $n-1$. We also provide a few extensions of this result.