{"id":"https://openalex.org/W1524380431","doi":"https://doi.org/10.1017/s096354831100037x","title":"Pl\u00fcnnecke's Inequality","display_name":"Pl\u00fcnnecke's Inequality","publication_year":2011,"publication_date":"2011-10-05","ids":{"openalex":"https://openalex.org/W1524380431","doi":"https://doi.org/10.1017/s096354831100037x","mag":"1524380431"},"language":"en","primary_location":{"id":"doi:10.1017/s096354831100037x","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s096354831100037x","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1101.2532","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023427657","display_name":"Giorgis Petridis","orcid":"https://orcid.org/0009-0007-5296-0547"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"GIORGIS PETRIDIS","raw_affiliation_strings":["Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail:","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5023427657"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.2619,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.72946522,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"20","issue":"6","first_page":"921","last_page":"938"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9771999716758728,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9768000245094299,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.7951170206069946},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7356384992599487},{"id":"https://openalex.org/keywords/cartesian-product","display_name":"Cartesian product","score":0.6620991230010986},{"id":"https://openalex.org/keywords/commutative-property","display_name":"Commutative property","score":0.6552149057388306},{"id":"https://openalex.org/keywords/inequality","display_name":"Inequality","score":0.6260942220687866},{"id":"https://openalex.org/keywords/novelty","display_name":"Novelty","score":0.5890470743179321},{"id":"https://openalex.org/keywords/key","display_name":"Key (lock)","score":0.5019686222076416},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.49489539861679077},{"id":"https://openalex.org/keywords/cartesian-coordinate-system","display_name":"Cartesian coordinate system","score":0.4571928083896637},{"id":"https://openalex.org/keywords/log-sum-inequality","display_name":"Log sum inequality","score":0.41784971952438354},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3887327015399933},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.13155236840248108},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09358879923820496},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.06178727746009827}],"concepts":[{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.7951170206069946},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7356384992599487},{"id":"https://openalex.org/C65236422","wikidata":"https://www.wikidata.org/wiki/Q173740","display_name":"Cartesian product","level":2,"score":0.6620991230010986},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.6552149057388306},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.6260942220687866},{"id":"https://openalex.org/C2778738651","wikidata":"https://www.wikidata.org/wiki/Q16546687","display_name":"Novelty","level":2,"score":0.5890470743179321},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.5019686222076416},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.49489539861679077},{"id":"https://openalex.org/C16038011","wikidata":"https://www.wikidata.org/wiki/Q62912","display_name":"Cartesian coordinate system","level":2,"score":0.4571928083896637},{"id":"https://openalex.org/C46979015","wikidata":"https://www.wikidata.org/wiki/Q6666859","display_name":"Log sum inequality","level":3,"score":0.41784971952438354},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3887327015399933},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.13155236840248108},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09358879923820496},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.06178727746009827},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0},{"id":"https://openalex.org/C27206212","wikidata":"https://www.wikidata.org/wiki/Q34178","display_name":"Theology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1017/s096354831100037x","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s096354831100037x","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1101.2532","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1101.2532","pdf_url":"https://arxiv.org/pdf/1101.2532","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":null},{"id":"mag:1524380431","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1101.2532","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.743.8657","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.743.8657","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1101.2532.pdf","raw_type":"text"},{"id":"doi:10.48550/arxiv.1101.2532","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1101.2532","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"},{"id":"mag:3111799105","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1101.2532","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1101.2532","pdf_url":"https://arxiv.org/pdf/1101.2532","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":null},"sustainable_development_goals":[{"score":0.6200000047683716,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W976211528","https://openalex.org/W1854241961","https://openalex.org/W1987254493","https://openalex.org/W2010258078","https://openalex.org/W2063292248","https://openalex.org/W2479604145","https://openalex.org/W2912209775"],"related_works":["https://openalex.org/W2127629487","https://openalex.org/W2320204860","https://openalex.org/W20105556","https://openalex.org/W3203914829","https://openalex.org/W2113835179","https://openalex.org/W2290305099","https://openalex.org/W2186859742","https://openalex.org/W2250110183","https://openalex.org/W2575486454","https://openalex.org/W2547088347","https://openalex.org/W912003938","https://openalex.org/W1732373389","https://openalex.org/W2968435267","https://openalex.org/W1497542966","https://openalex.org/W1981909532","https://openalex.org/W2999223002","https://openalex.org/W1516711503","https://openalex.org/W2949884361","https://openalex.org/W1705463008","https://openalex.org/W2060782772"],"abstract_inverted_index":{"Pl\u00fcnnecke's":[0],"inequality":[1,53,93],"is":[2,29,33,76],"a":[3,23,88],"standard":[4],"tool":[5],"for":[6,61,66,91],"obtaining":[7],"estimates":[8],"on":[9],"the":[10,31,49,52,77,92],"cardinality":[11],"of":[12,44,51,79],"sumsets":[13],"and":[14,54],"has":[15],"many":[16],"applications":[17],"in":[18,73],"additive":[19],"combinatorics.":[20],"We":[21,46],"present":[22],"new":[24],"proof.":[25],"The":[26],"main":[27],"novelty":[28],"that":[30,56],"proof":[32],"completed":[34],"with":[35],"no":[36],"reference":[37],"to":[38,94],"Menger's":[39],"theorem":[40],"or":[41],"Cartesian":[42],"products":[43],"graphs.":[45,69,84],"also":[47],"investigate":[48],"sharpness":[50],"show":[55],"it":[57],"can":[58],"be":[59,95],"sharp":[60],"arbitrarily":[62,80],"long,":[63],"but":[64],"not":[65],"infinite":[67],"commutative":[68,83],"A":[70],"key":[71],"step":[72],"our":[74],"investigation":[75],"construction":[78],"long":[81],"regular":[82],"Lastly":[85],"we":[86],"prove":[87],"necessary":[89],"condition":[90],"attained.":[96]},"counts_by_year":[{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2}],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2025-10-10T00:00:00"}
