On a Conjecture of Fox and Kleitman on the Degree of Regularity of a Certain Linear Equation

Abstract

Fox and Kleitman proved in 2006 that for any positive integer b, the 2n-variable equation $x_1+\cdots +x_n - x_{n+1}- \cdots - x_{2n} \ = \ b$ is not 2n-regular. Moreover, they conjectured the existence of an integer $b_n \ge 1$ such that for $b=b_n$, this equation is $(2n-1)$-regular. In this note, we settle the first nontrivial case of the conjecture, namely for $n=2$, and we propose a slight refinement of it.