Resolving a Conjecture on Degree of Regularity of Linear Homogeneous Equations

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 Radoićič conjectured that the equation x1 +2x2 + · · ·+2xn−1−2xn = 0, for any n > 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 Radoićič’s family of equations indeed have a degree of regularity of n− 1. We also prove a few extensions of this result.