Resolving a Conjecture on Degree of Regularity of Linear Homogeneous 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 $n\ge 2$, 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.

8 pages

Cited by in corpus (1)

• Establishing Conditions on the Degree of Regularity of Linear Homogeneous Equations