{"id":"https://openalex.org/W7215748603","doi":"https://doi.org/10.48550/arxiv.2610.00375","title":"Optimal discrepancy for triangles","display_name":"Optimal discrepancy for triangles","publication_year":2026,"publication_date":"2026-09-30","ids":{"openalex":"https://openalex.org/W7215748603","doi":"https://doi.org/10.48550/arxiv.2610.00375"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.00375","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00375","pdf_url":"https://arxiv.org/pdf/2610.00375","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.00375","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5070458758","display_name":"Gleb Smirnov","orcid":"https://orcid.org/0009-0007-3097-952X","observed_orcids":["https://orcid.org/0009-0007-3097-952X"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Smirnov, Gleb","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5070458758"],"corresponding_institution_ids":[],"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9369999766349792,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9369999766349792,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.012500000186264515,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.011500000022351742,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/discrepancy-theory","display_name":"discrepancy theory","score":0.6840000152587891},{"id":"https://openalex.org/keywords/2-coloring","display_name":"two-coloring","score":0.7979999780654907},{"id":"https://openalex.org/keywords/triangles","display_name":"triangles","score":0.6779999732971191},{"id":"https://openalex.org/keywords/planar-point-sets","display_name":"planar point sets","score":0.7179999947547913},{"id":"https://openalex.org/keywords/geometric-discrepancy","display_name":"geometric discrepancy","score":0.5899999737739563},{"id":"https://openalex.org/keywords/lower-bounds","display_name":"lower bounds","score":0.45500001311302185}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6064000129699707},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5199999809265137},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5006999969482422},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.45419999957084656},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.37389999628067017},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3197999894618988},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.2809999883174896},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.27070000767707825},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.23420000076293945},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.23420000076293945}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.00375","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00375","pdf_url":"https://arxiv.org/pdf/2610.00375","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.00375","doi":"https://doi.org/10.48550/arxiv.2610.00375","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.00375","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:2610.00375","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00375","pdf_url":"https://arxiv.org/pdf/2610.00375","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":null,"license_id":null,"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/W7215748603.pdf","grobid_xml":"https://content.openalex.org/works/W7215748603.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7220959476","https://openalex.org/W7214658798","https://openalex.org/W1993401990","https://openalex.org/W7095949263","https://openalex.org/W7204169707","https://openalex.org/W7204784550","https://openalex.org/W2397108213","https://openalex.org/W2070611841","https://openalex.org/W2107001016","https://openalex.org/W2163000644"],"abstract_inverted_index":{"We":[0],"prove":[1],"that":[2],"every":[3],"$n$-point":[4],"set":[5],"in":[6],"the":[7,20,27],"plane":[8],"has":[9],"a":[10],"two-coloring":[11],"with":[12],"discrepancy":[13],"$O(n^{1/4})$":[14],"for":[15],"all":[16],"triangles.":[17],"This":[18],"improves":[19],"previous":[21],"bound":[22],"$O(n^{1/4}\\sqrt{\\log":[23],"n})$":[24],"and":[25],"matches":[26],"classical":[28],"lower":[29],"bound.":[30]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-03T00:00:00"}