On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1

For integers k, n, c with k, n ≥ 1, and c ≥ 0, the n-color weak Rado number WRk (n, c) is defined as the least integer N, if it exists, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution x1 ,..., xk, xk+1 in that interval to the equation x1 + x2 +···...

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Detalles Bibliográficos
Autores: Boza Prieto, Luis, 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:2019
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/135999
Acceso en línea:https://hdl.handle.net/11441/135999
https://doi.org/10.1080/10586458.2017.1382403
Access Level:acceso abierto
Palabra clave:Schur numbers
Sum-free sets
Weak Schur numbers
Weakly sum-free sets
Rado numbers
Weak Rado numbers
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spelling On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1Boza Prieto, LuisMarín Sánchez, Juan ManuelRevuelta Marchena, María PastoraSanz Domínguez, María IsabelSchur numbersSum-free setsWeak Schur numbersWeakly sum-free setsRado numbersWeak Rado numbersFor integers k, n, c with k, n ≥ 1, and c ≥ 0, the n-color weak Rado number WRk (n, c) is defined as the least integer N, if it exists, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution x1 ,..., xk, xk+1 in that interval to the equation x1 + x2 +···+ xk + c = xk+1 , with xi = xj , when i = j. If no such N exists, then WRk (n, c) is defined as infinite. In this paper, we determine the exact value of some of these numbers for n = 2 and n = 3, namely WR3 (2, c) = 5c + 24, WR4(2, c) = 6c + 52 for all c ≥ 0 and WR2 (3, c) = 13c + 22 for all c > 0. Our method consists in translating the problem into a Boolean satisfiability problem, which can then be handled by a SAT solver or by backtrack programming in the language C.Taylor and FrancisMatemática Aplicada IFQM-164: Matemática Discreta: Teoría de Grafos y Geometría Computacional FQM-240: Invariantes en Teoría de Grafos y Optimización2019info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/135999https://doi.org/10.1080/10586458.2017.1382403reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésExperimental Mathematics, 28 (2), 194-208.https://www.tandfonline.com/doi/full/10.1080/10586458.2017.1382403info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1359992026-06-17T12:51:07Z
dc.title.none.fl_str_mv On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1
title On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1
spellingShingle On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1
Boza Prieto, Luis
Schur numbers
Sum-free sets
Weak Schur numbers
Weakly sum-free sets
Rado numbers
Weak Rado numbers
title_short On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1
title_full On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1
title_fullStr On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1
title_full_unstemmed On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1
title_sort On the n-Color Weak Rado Numbers for the Equation x1 + x2 + ··· + xk + c = xk +1
dc.creator.none.fl_str_mv Boza Prieto, Luis
Marín Sánchez, Juan Manuel
Revuelta Marchena, María Pastora
Sanz Domínguez, María Isabel
author Boza Prieto, Luis
author_facet Boza Prieto, Luis
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
FQM-240: Invariantes en Teoría de Grafos y Optimización
dc.subject.none.fl_str_mv Schur numbers
Sum-free sets
Weak Schur numbers
Weakly sum-free sets
Rado numbers
Weak Rado numbers
topic Schur numbers
Sum-free sets
Weak Schur numbers
Weakly sum-free sets
Rado numbers
Weak Rado numbers
description For integers k, n, c with k, n ≥ 1, and c ≥ 0, the n-color weak Rado number WRk (n, c) is defined as the least integer N, if it exists, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution x1 ,..., xk, xk+1 in that interval to the equation x1 + x2 +···+ xk + c = xk+1 , with xi = xj , when i = j. If no such N exists, then WRk (n, c) is defined as infinite. In this paper, we determine the exact value of some of these numbers for n = 2 and n = 3, namely WR3 (2, c) = 5c + 24, WR4(2, c) = 6c + 52 for all c ≥ 0 and WR2 (3, c) = 13c + 22 for all c > 0. Our method consists in translating the problem into a Boolean satisfiability problem, which can then be handled by a SAT solver or by backtrack programming in the language C.
publishDate 2019
dc.date.none.fl_str_mv 2019
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/11441/135999
https://doi.org/10.1080/10586458.2017.1382403
url https://hdl.handle.net/11441/135999
https://doi.org/10.1080/10586458.2017.1382403
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Experimental Mathematics, 28 (2), 194-208.
https://www.tandfonline.com/doi/full/10.1080/10586458.2017.1382403
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 Taylor and Francis
publisher.none.fl_str_mv Taylor and Francis
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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