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 ∑k i=1 (xi − yi) = 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. Wh...
| Autores: | , , , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2018 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/136014 |
| Acceso en línea: | https://hdl.handle.net/11441/136014 https://doi.org/10.1016/j.disc.2017.08.040 |
| Access Level: | acceso abierto |
| Palabra clave: | Partition regularity Degree of regularity Monochromatic solution Discrete derivative |
| Sumario: | 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 ∑k i=1 (xi − yi) = 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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