Weak Schur numbers and the search for G.W. Walker’s lost partitions

A set A of integers is weakly sum-free if it contains no three distinct elements x, y, z such that x + y = z. Given k ≥ 1, let WS(k) denote the largest integer n for which {1, . . . , n} admits a partition into k weakly sum-free subsets. In 1952, G.W. Walker claimed the value WS(5) = 196, without pr...

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Detalles Bibliográficos
Autores: Eliahou, Shalom, Marín Sánchez, Juan Manuel, Revuelta Marchena, María Pastora, Sanz Domínguez, María Isabel
Tipo de recurso: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2012
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/136600
Acceso en línea:https://hdl.handle.net/11441/136600
https://doi.org/0.1016/j.camwa.2011.11.006
Access Level:acceso abierto
Palabra clave:Schur numbers
Sum-free sets
Weakly sum-free sets
Boolean variables
SAT problem
SAT-solvers
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spelling Weak Schur numbers and the search for G.W. Walker’s lost partitionsEliahou, ShalomMarín Sánchez, Juan ManuelRevuelta Marchena, María PastoraSanz Domínguez, María IsabelSchur numbersSum-free setsWeakly sum-free setsBoolean variablesSAT problemSAT-solversA set A of integers is weakly sum-free if it contains no three distinct elements x, y, z such that x + y = z. Given k ≥ 1, let WS(k) denote the largest integer n for which {1, . . . , n} admits a partition into k weakly sum-free subsets. In 1952, G.W. Walker claimed the value WS(5) = 196, without proof. Here we show WS(5) ≥ 196, by constructing a partition of {1, . . . , 196} of the required type. It remains as an open problem to prove the equality. With an analogous construction for k = 6, we obtain WS(6) ≥ 572. Our approach involves translating the construction problem into a Boolean satisfiability problem, which can then be handled by a SAT solver.ElsevierMatemática Aplicada IFQM-164: Matemática Discreta: Teoría de Grafos y Geometría Computacional2012info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/136600https://doi.org/0.1016/j.camwa.2011.11.006reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésComputers and Mathematics with Applications, 63 (1), 175-182.https://www.sciencedirect.com/science/article/pii/S0898122111009722?via%3Dihubinfo:eu-repo/semantics/openAccessoai:idus.us.es:11441/1366002026-06-17T12:51:07Z
dc.title.none.fl_str_mv Weak Schur numbers and the search for G.W. Walker’s lost partitions
title Weak Schur numbers and the search for G.W. Walker’s lost partitions
spellingShingle Weak Schur numbers and the search for G.W. Walker’s lost partitions
Eliahou, Shalom
Schur numbers
Sum-free sets
Weakly sum-free sets
Boolean variables
SAT problem
SAT-solvers
title_short Weak Schur numbers and the search for G.W. Walker’s lost partitions
title_full Weak Schur numbers and the search for G.W. Walker’s lost partitions
title_fullStr Weak Schur numbers and the search for G.W. Walker’s lost partitions
title_full_unstemmed Weak Schur numbers and the search for G.W. Walker’s lost partitions
title_sort Weak Schur numbers and the search for G.W. Walker’s lost partitions
dc.creator.none.fl_str_mv Eliahou, Shalom
Marín Sánchez, Juan Manuel
Revuelta Marchena, María Pastora
Sanz Domínguez, María Isabel
author Eliahou, Shalom
author_facet Eliahou, Shalom
Marín Sánchez, Juan Manuel
Revuelta Marchena, María Pastora
Sanz Domínguez, María Isabel
author_role author
author2 Marín Sánchez, Juan Manuel
Revuelta Marchena, María Pastora
Sanz Domínguez, María Isabel
author2_role author
author
author
dc.contributor.none.fl_str_mv Matemática Aplicada I
FQM-164: Matemática Discreta: Teoría de Grafos y Geometría Computacional
dc.subject.none.fl_str_mv Schur numbers
Sum-free sets
Weakly sum-free sets
Boolean variables
SAT problem
SAT-solvers
topic Schur numbers
Sum-free sets
Weakly sum-free sets
Boolean variables
SAT problem
SAT-solvers
description A set A of integers is weakly sum-free if it contains no three distinct elements x, y, z such that x + y = z. Given k ≥ 1, let WS(k) denote the largest integer n for which {1, . . . , n} admits a partition into k weakly sum-free subsets. In 1952, G.W. Walker claimed the value WS(5) = 196, without proof. Here we show WS(5) ≥ 196, by constructing a partition of {1, . . . , 196} of the required type. It remains as an open problem to prove the equality. With an analogous construction for k = 6, we obtain WS(6) ≥ 572. Our approach involves translating the construction problem into a Boolean satisfiability problem, which can then be handled by a SAT solver.
publishDate 2012
dc.date.none.fl_str_mv 2012
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/11441/136600
https://doi.org/0.1016/j.camwa.2011.11.006
url https://hdl.handle.net/11441/136600
https://doi.org/0.1016/j.camwa.2011.11.006
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Computers and Mathematics with Applications, 63 (1), 175-182.
https://www.sciencedirect.com/science/article/pii/S0898122111009722?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 15.300719

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