{"id":"https://openalex.org/W4383962004","doi":"https://doi.org/10.48550/arxiv.2307.03531","title":"A Note On The Cross-Sperner Families","display_name":"A Note On The Cross-Sperner Families","publication_year":2023,"publication_date":"2023-07-07","ids":{"openalex":"https://openalex.org/W4383962004","doi":"https://doi.org/10.48550/arxiv.2307.03531"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2307.03531","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2307.03531","pdf_url":"https://arxiv.org/pdf/2307.03531","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,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"parent_works":[],"lineage":["https://openalex.org/W4383962004"],"child_works_count":0,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2307.03531","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5014136523","display_name":"Junyao Pan","orcid":"https://orcid.org/0000-0002-0214-3860","observed_orcids":["https://orcid.org/0000-0002-0214-3860"]},"institutions":[{"id":"https://openalex.org/I4405259936","display_name":"Wuxi University","ror":"https://ror.org/02yq4ac41","country_code":"CN","type":"education","lineage":["https://openalex.org/I4405259936"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Pan, Junyao","raw_affiliation_strings":["School of Sciences , Wuxi University , Wu'xi , Jiangsu , 214105 People's Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Sciences , Wuxi University , Wu'xi , Jiangsu , 214105 People's Republic of China","institution_ids":["https://openalex.org/I4405259936"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5014136523"],"corresponding_institution_ids":["https://openalex.org/I4405259936"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9922000169754028,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9922000169754028,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.007000000216066837,"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/T10374","display_name":"Advanced Graph Theory Research","score":9.999999747378752e-05,"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/extremal-set-theory","display_name":"extremal set theory","score":0.7900000214576721},{"id":"https://openalex.org/keywords/intersecting-families","display_name":"intersecting families","score":0.652999997138977},{"id":"https://openalex.org/keywords/set-intersection","display_name":"set intersection","score":0.5669999718666077}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8394365906715393},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.4377821981906891},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3972187042236328}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2307.03531","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2307.03531","pdf_url":"https://arxiv.org/pdf/2307.03531","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,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2307.03531","doi":"https://doi.org/10.48550/arxiv.2307.03531","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2307.03531","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,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2307.03531","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2307.03531","pdf_url":"https://arxiv.org/pdf/2307.03531","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,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4383962004.pdf","grobid_xml":"https://content.openalex.org/works/W4383962004.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W4394934440","https://openalex.org/W4402381134"],"related_works":["https://openalex.org/W2999629182","https://openalex.org/W7097541501","https://openalex.org/W4394934440","https://openalex.org/W1980907826","https://openalex.org/W4404987135","https://openalex.org/W2034065916","https://openalex.org/W7167602861","https://openalex.org/W1652501426","https://openalex.org/W2950933589","https://openalex.org/W6999558427"],"abstract_inverted_index":{"Let":[0],"$(\\mathcal{F},\\mathcal{G})$":[1,24],"be":[2],"a":[3,27],"pair":[4],"of":[5,7],"families":[6],"$[n]$,":[8],"where":[9,41],"$[n]=\\{1,2,...,n\\}$.":[10],"If":[11],"$A\\not\\subset":[12],"B$":[13],"and":[14,21,32,63],"$B\\not\\subset":[15],"A$":[16],"hold":[17],"for":[18,51],"all":[19,52],"$A\\in\\mathcal{F}$":[20],"$B\\in\\mathcal{G}$,":[22],"then":[23],"is":[25],"called":[26],"Cross-Sperner":[28],"pair.":[29],"P.":[30,61],"Frankl":[31,62],"Jian":[33,64],"Wang":[34],"introduced":[35],"the":[36],"extremal":[37],"problem":[38,58],"that":[39,49],"$m(n)={\\rm{max}}\\{|\\mathcal{I}(\\mathcal{F},\\mathcal{G})|:\\mathcal{F},\\mathcal{G}\\subset2^{[n]}~{\\rm{are~cross}}$-${\\rm{sperner}}\\}$,":[40],"$\\mathcal{I}(\\mathcal{F},\\mathcal{G})=\\{A\\cap":[42],"B:A\\in\\mathcal{F},B\\in\\mathcal{G}\\}$.":[43],"In":[44],"this":[45],"note,":[46],"we":[47],"prove":[48],"$m(n)=2^n-2^{\\lfloor\\frac{n}{2}\\rfloor}-2^{\\lceil\\frac{n}{2}\\rceil}+1$":[50],"$n>1$.":[53],"This":[54],"solves":[55],"an":[56],"open":[57],"proposed":[59],"by":[60],"Wang.":[65]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2023-07-12T00:00:00"}
