Establishing Conditions on the Degree of Regularity of Linear Homogeneous Equations

In 1933, Rado conjectured that for any positive integer n, there is always a linear homogeneous equation with degree of regularity n. In proving this conjecture, Alexeev and Tsimerman, and independently Golowich, found that some equations in n variables have degree of regularity n-1 for any value of n. Their work left many questions as to how and which other properties of equations are closely tied to the degree of regularity, and if there is a simpler or more effective way of thinking about it. In this paper, we answer some of these questions, prove that various families of linear homogeneous equations in n variables have degree of regularity n-1, and establish some conditions under which this property holds.