Equation-regular sets and the Fox–Kleitman conjecture
Résumé
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 \sum_{i=1}^k (x_i − y_i) = b is (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.
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