{"id":"https://openalex.org/W7161940455","doi":"https://doi.org/10.1007/s00493-026-00216-z","title":"A Universal Threshold for Geometric Embeddings of Trees","display_name":"A Universal Threshold for Geometric Embeddings of Trees","publication_year":2026,"publication_date":"2026-05-21","ids":{"openalex":"https://openalex.org/W7161940455","doi":"https://doi.org/10.1007/s00493-026-00216-z"},"language":"en","primary_location":{"id":"doi:10.1007/s00493-026-00216-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00493-026-00216-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00493-026-00216-z.pdf","source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorica","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00493-026-00216-z.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051371597","display_name":"Dylan J. Altschuler","orcid":null},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Dylan J. Altschuler","raw_affiliation_strings":["Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, USA","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5026140601","display_name":"Pandelis Dodos","orcid":"https://orcid.org/0000-0002-7231-5074"},"institutions":[{"id":"https://openalex.org/I200777214","display_name":"National and Kapodistrian University of Athens","ror":"https://ror.org/04gnjpq42","country_code":"GR","type":"education","lineage":["https://openalex.org/I200777214"]}],"countries":["GR"],"is_corresponding":true,"raw_author_name":"Pandelis Dodos","raw_affiliation_strings":["Department of Mathematics, University of Athens, Athens, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Athens, Athens, Greece","institution_ids":["https://openalex.org/I200777214"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5136635364","display_name":"Konstantin Tikhomirov","orcid":null},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Konstantin Tikhomirov","raw_affiliation_strings":["Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, USA","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5090518786","display_name":"Konstantinos Tyros","orcid":"https://orcid.org/0000-0002-8890-7074"},"institutions":[{"id":"https://openalex.org/I200777214","display_name":"National and Kapodistrian University of Athens","ror":"https://ror.org/04gnjpq42","country_code":"GR","type":"education","lineage":["https://openalex.org/I200777214"]}],"countries":["GR"],"is_corresponding":true,"raw_author_name":"Konstantinos Tyros","raw_affiliation_strings":["Department of Mathematics, University of Athens, Athens, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Athens, Athens, Greece","institution_ids":["https://openalex.org/I200777214"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5026140601","https://openalex.org/A5051371597","https://openalex.org/A5090518786","https://openalex.org/A5136635364"],"corresponding_institution_ids":["https://openalex.org/I200777214","https://openalex.org/I74973139"],"apc_list":{"value":3290,"currency":"USD","value_usd":3290},"apc_paid":{"value":3290,"currency":"USD","value_usd":3290},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.59566894,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"46","issue":"3","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.5455999970436096,"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"}},"topics":[{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.5455999970436096,"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":0.11550000309944153,"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"}},{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.06430000066757202,"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/dimension","display_name":"Dimension (graph theory)","score":0.6355000138282776},{"id":"https://openalex.org/keywords/normed-vector-space","display_name":"Normed vector space","score":0.5246000289916992},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.5170999765396118},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.48579999804496765},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.46480000019073486},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.445499986410141}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.916100025177002},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7271999716758728},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6355000138282776},{"id":"https://openalex.org/C171739935","wikidata":"https://www.wikidata.org/wiki/Q726210","display_name":"Normed vector space","level":2,"score":0.5246000289916992},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5170999765396118},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48969998955726624},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.48579999804496765},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.46480000019073486},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.445499986410141},{"id":"https://openalex.org/C162860070","wikidata":"https://www.wikidata.org/wiki/Q142140","display_name":"Finitely-generated abelian group","level":2,"score":0.2953000068664551},{"id":"https://openalex.org/C2778114796","wikidata":"https://www.wikidata.org/wiki/Q7575194","display_name":"Spectral gap","level":2,"score":0.2825999855995178},{"id":"https://openalex.org/C64331007","wikidata":"https://www.wikidata.org/wiki/Q831672","display_name":"Spanning tree","level":2,"score":0.25870001316070557},{"id":"https://openalex.org/C154547637","wikidata":"https://www.wikidata.org/wiki/Q776602","display_name":"Expander graph","level":3,"score":0.2549999952316284}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00493-026-00216-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00493-026-00216-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00493-026-00216-z.pdf","source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorica","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s00493-026-00216-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00493-026-00216-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00493-026-00216-z.pdf","source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorica","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8599871579","display_name":"Random Matrices and Functional Inequalities on Spaces of Graphs","funder_award_id":"2331037","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320318628","display_name":"Hellenic Academic Libraries Link","ror":null},{"id":"https://openalex.org/F4320320300","display_name":"European Commission","ror":"https://ror.org/00k4n6c32"},{"id":"https://openalex.org/F4320323149","display_name":"National and Kapodistrian University of Athens","ror":"https://ror.org/04gnjpq42"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7161940455.pdf","grobid_xml":"https://content.openalex.org/works/W7161940455.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W1508659029","https://openalex.org/W1540940340","https://openalex.org/W1981853164","https://openalex.org/W1994464108","https://openalex.org/W2001505356","https://openalex.org/W2019029270","https://openalex.org/W2028203362","https://openalex.org/W2028712970","https://openalex.org/W2051997748","https://openalex.org/W2066864620","https://openalex.org/W2132455346","https://openalex.org/W2983923309","https://openalex.org/W3138914377","https://openalex.org/W4387316511","https://openalex.org/W4413888283","https://openalex.org/W4414688519"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"A":[1],"graph":[2],"$$G=(V,E)$$":[3],"G":[4],"=":[5],"(":[6,42,47],"V":[7,29,77],",":[8,62,67,74,98,144],"E":[9,66],")":[10,44,49],"is":[11,21,81,124,188],"geometrically":[12],"embeddable":[13,125],"into":[14,126,150],"a":[15,22,191],"normed":[16,128,152],"space":[17,129,153],"X":[18,31,51],"when":[19],"there":[20],"mapping":[23],"$$\\zeta":[24],":V\\rightarrow":[25],"X$$":[26],"\u03b6":[27,41,46],":":[28],"\u2192":[30],"such":[32],"that":[33],"$$\\Vert":[34],"\\zeta":[35],"(v)-\\zeta":[36],"(w)\\Vert":[37],"_X\\leqslant":[38],"1$$":[39],"\u2016":[40,50],"v":[43,61,73],"-":[45],"w":[48,63,75],"\u2a7d":[52],"1":[53,162],"if":[54,57],"and":[55,99,145,175],"only":[56],"$$\\{v,w\\}\\in":[58],"E$$":[59],"{":[60],"}":[64],"\u2208":[65,76],"for":[68,86],"all":[69],"distinct":[70],"$$v,w\\in":[71],"V$$":[72],".":[78,111,169],"Our":[79,186],"result":[80,187],"the":[82,87,197],"following":[83],"universal":[84],"threshold":[85],"embeddability":[88],"of":[89,107,116,130,154],"trees.":[90],"Let":[91],"$$\\Delta":[92,108,121],"\\geqslant":[93],"3$$":[94],"\u0394":[95,110,123],"\u2a7e":[96],"3":[97],"let":[100],"N":[101,113,140,143,165,168],"be":[102,181],"sufficiently":[103],"large":[104],"in":[105,183],"terms":[106],"$$":[109,122],"Every":[112],"\u2013vertex":[114],"tree":[115],"maximal":[117],"degree":[118],"at":[119,132],"most":[120],"any":[127,151],"dimension":[131,155],"least":[133],"$$64\\,\\frac{\\log":[134],"N}{\\log":[135,159],"\\log":[136,160],"N}$$":[137,161],"64":[138],"log":[139,141,142,164,166,167],"complete":[146],"trees":[147],"are":[148,178],"non-embeddable":[149,182],"less":[156],"than":[157],"$$\\frac{1}{2}\\,\\frac{\\log":[158],"2":[163],"In":[170],"striking":[171],"contrast,":[172],"spectral":[173],"expanders":[174],"random":[176],"graphs":[177],"known":[179],"to":[180],"sublogarithmic":[184],"dimension.":[185],"based":[189],"on":[190,200],"randomized":[192],"embedding":[193],"whose":[194],"analysis":[195],"utilizes":[196],"recent":[198],"breakthroughs":[199],"Bourgain\u2019s":[201],"slicing":[202],"problem.":[203]},"counts_by_year":[],"updated_date":"2026-08-22T07:34:49.880490","created_date":"2026-05-22T00:00:00"}