{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-DMRDIX09/21d8637e-679b-4853-b5a2-f898c8cd63c6/PDF","dcterms:extent":"245 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-DMRDIX09/0bcbfbd5-557d-4837-96c3-feaea1f5470a/TEXT","dcterms:extent":"17 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-DMRDIX09","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2020","dc:creator":"Budden, Mark","dc:format":[{"@xml:lang":"sl","#text":"letnik:18"},{"@xml:lang":"sl","#text":"številka:2"},{"@xml:lang":"sl","#text":"str. 281-288"}],"dc:identifier":["ISSN:1855-3966","COBISSID_HOST:41166339","URN:URN:NBN:SI:doc-DMRDIX09"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"en","#text":"anti-Ramsey numbers"},{"@xml:lang":"sl","#text":"anti-Ramseyeva števila"},{"@xml:lang":"en","#text":"Gallai colorings"},{"@xml:lang":"sl","#text":"Gallaieva barvanja"},{"@xml:lang":"sl","#text":"mavrični trikotniki"},{"@xml:lang":"en","#text":"rainbow triangles"},{"@xml:lang":"en","#text":"Schur numbers"},{"@xml:lang":"sl","#text":"Schurova števila"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Schur numbers involving rainbow colorings|"},"dc:description":[{"@xml:lang":"sl","#text":"In this paper, we introduce two different generalizations of Schur numbers that involve rainbow colorings. Motivated by well-known generalizations of Ramsey numbers, we first define the rainbow Schur number ?$RS(n)$? to be the minimum number of colors needed such that every coloring of ?$\\{1, 2, \\dots , n\\}$?, in which all available colors are used, contains a rainbow solution to ?$a+b=c$?. It is shown that ?$$RS(n)=\\lfloor \\log_2(n)\\rfloor + 2, \\quad \\text{for all} \\; n \\ge 3.$$? Second, we consider the Gallai-Schur number ?$GS(n)$?, defined to be the least natural number such that every ?$n$?-coloring of ?$\\{1, 2,\\dots, GS(n)\\}$? that lacks rainbow solutions to the equation ?$a+b=c$? necessarily contains a monochromatic solution to this equation. By connecting this number with the ?$n$?-color Gallai-Ramsey number for triangles, it is shown that for all ?$n \\ge 3$?, ?$$ GS(n) = \\begin{cases} 5^k & \\text{if} \\; n=2k \\\\ 2 \\cdot 5^k & \\text{if} \\; n=2k+1. \\\\ \\end{cases}$$?"},{"@xml:lang":"sl","#text":"V članku vpeljemo dve posplošitvi Schurovih števil, povezani z mavričnimi barvanji. Spodbujeni z dobro znanimi posplošitvami Ramseyevih števil, najprej definiramo mavrično Schurovo število ?$RS(n)$? kot najmanjše število barv, potrebnih za to, da vsako barvanje množice ?$\\{1, 2, \\dots , n\\}$?, v katerem so uporabljene vse barve, ki so na voljo, vsebuje mavrično rešitev enačbe ?$a+b=c$?. Dokažemo, da velja ?$$RS(n)=\\lfloor \\log_2(n)\\rfloor + 2, \\quad \\text{for all} \\; n \\ge 3.$$? Kot drugo, obravnavamo Gallai-Schurovo število ?$GS(n)$?, definirano kot najmanjše naravno število, pri katerem velja, da vsako ?$n$?-barvanje množice ?$\\{1, 2,\\dots, GS(n)\\}$?, ki ne premore mavričnih rešitev enačbe ?$a+b=c$?, neizogibno vsebuje enobarvno rešitev te enačbe. To število povežemo z ?$n$?-barvnim Gallai-Ramseyevim številom za trikotnike in dokažemo, da za vse ?$n \\ge 3$? velja ?$$ GS(n) = \\begin{cases} 5^k & \\text{if} \\; n=2k \\\\ 2 \\cdot 5^k & \\text{if} \\; n=2k+1. \\\\ \\end{cases}$$?"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-DMRDIX09","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-DMRDIX09"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-DMRDIX09/21d8637e-679b-4853-b5a2-f898c8cd63c6/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-DMRDIX09/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-DMRDIX09"}}}}