Equation-regular sets and the Fox-Kleitman conjecture

Given k≥1, the Fox-Kleitman conjecture from 2006 states that there exists a nonzero integer b such that the 2k-variable linear Diophantine equation ∑i=1k(xi−yi)=bis (2k−1)-regular. This is best possible, since Fox and Kleitman showed that for all b≥1, this equation is not 2k-regular. While the conjecture has recently been settled for all k≥2, here we focus on the case k=3 and determine the degree of regularity of the corresponding equation for all b≥1. In particular, this independently confirms the conjecture for k=3. We also briefly discuss the case k=4.