  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.
