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Journal Articles Discrete Mathematics Year : 2018

Equation-regular sets and the Fox–Kleitman conjecture

S.D. Adhikari
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L. Boza
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M.P. Revuelta
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M.I. Sanz
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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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Dates and versions

hal-04147403 , version 1 (30-06-2023)



S.D. Adhikari, L. Boza, Shalom Eliahou, M.P. Revuelta, M.I. Sanz. Equation-regular sets and the Fox–Kleitman conjecture. Discrete Mathematics, 2018, 341 (2), pp.287-298. ⟨10.1016/j.disc.2017.08.040⟩. ⟨hal-04147403⟩
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