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...

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Detalles Bibliográficos
Autores: Adhikari, S. D., Boza Prieto, Luis, Eliahou, Shalom, Revuelta Marchena, María Pastora, Sanz Domínguez, María Isabel
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
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spelling Equation-regular sets and the Fox–Kleitman conjectureAdhikari, S. D.Boza Prieto, LuisEliahou, ShalomRevuelta Marchena, María PastoraSanz Domínguez, María IsabelPartition regularityDegree of regularityMonochromatic solutionDiscrete derivativeGiven 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.ElsevierMatemática Aplicada IFQM-164: Matemática Discreta: Teoría de Grafos y Geometría Computacional FQM-240: Invariantes en Teoría de Grafos y Optimización2018info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/136014https://doi.org/10.1016/j.disc.2017.08.040reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésDiscrete Mathematics, 341 (2), 287-298.https://www.sciencedirect.com/science/article/pii/S0012365X17302984?via%3Dihubinfo:eu-repo/semantics/openAccessoai:idus.us.es:11441/1360142026-06-17T12:51:07Z
dc.title.none.fl_str_mv Equation-regular sets and the Fox–Kleitman conjecture
title Equation-regular sets and the Fox–Kleitman conjecture
spellingShingle Equation-regular sets and the Fox–Kleitman conjecture
Adhikari, S. D.
Partition regularity
Degree of regularity
Monochromatic solution
Discrete derivative
title_short Equation-regular sets and the Fox–Kleitman conjecture
title_full Equation-regular sets and the Fox–Kleitman conjecture
title_fullStr Equation-regular sets and the Fox–Kleitman conjecture
title_full_unstemmed Equation-regular sets and the Fox–Kleitman conjecture
title_sort Equation-regular sets and the Fox–Kleitman conjecture
dc.creator.none.fl_str_mv Adhikari, S. D.
Boza Prieto, Luis
Eliahou, Shalom
Revuelta Marchena, María Pastora
Sanz Domínguez, María Isabel
author Adhikari, S. D.
author_facet Adhikari, S. D.
Boza Prieto, Luis
Eliahou, Shalom
Revuelta Marchena, María Pastora
Sanz Domínguez, María Isabel
author_role author
author2 Boza Prieto, Luis
Eliahou, Shalom
Revuelta Marchena, María Pastora
Sanz Domínguez, María Isabel
author2_role author
author
author
author
dc.contributor.none.fl_str_mv Matemática Aplicada I
FQM-164: Matemática Discreta: Teoría de Grafos y Geometría Computacional
FQM-240: Invariantes en Teoría de Grafos y Optimización
dc.subject.none.fl_str_mv Partition regularity
Degree of regularity
Monochromatic solution
Discrete derivative
topic Partition regularity
Degree of regularity
Monochromatic solution
Discrete derivative
description 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.
publishDate 2018
dc.date.none.fl_str_mv 2018
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/submittedVersion
format article
status_str submittedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/136014
https://doi.org/10.1016/j.disc.2017.08.040
url https://hdl.handle.net/11441/136014
https://doi.org/10.1016/j.disc.2017.08.040
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Discrete Mathematics, 341 (2), 287-298.
https://www.sciencedirect.com/science/article/pii/S0012365X17302984?via%3Dihub
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.301603

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