Degree of Regularity of Linear Homogeneous Equations

Abstract

We define a linear homogeneous equation to be strongly r-regular if, when a finite number of inequalities is added to the equation, the system of the equation and inequalities is still r-regular. In this paper, we show that, if a linear homogeneous equation is r-regular, then it is strongly r-regular. In 2009, Alexeev and Tsimerman introduced a family of equations, each of which is (n-1)-regular but not n-regular, verifying a conjecture of Rado from 1933. These equations are actually strongly (n-1)-regular as an immediate corollary of our results.

arXiv comment7 pages. This is an update of our previous version that proves a stronger result

References in corpus (1)

• Equations resolving a conjecture of Rado on partition regularity