Resolving a Conjecture on Degree of Regularity of Linear Homogeneous Equations

الأصل · EN

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 x₁ + 2x₂ + + 2ⁿ⁻²xₙ₋₁ - 2ⁿ⁻¹xₙ = 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 Radoicic's family of equations indeed have a degree of regularity of n-1. We also provide a few extensions of this result.