{"id":"https://openalex.org/W7225310940","doi":"https://doi.org/10.5281/zenodo.23972681","title":"A quadratic lower bound for intervals in the Hales\u2013Jewett theorem","display_name":"A quadratic lower bound for intervals in the Hales\u2013Jewett theorem","publication_year":2026,"publication_date":"2026-11-05","ids":{"openalex":"https://openalex.org/W7225310940","doi":"https://doi.org/10.5281/zenodo.23972681"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23972681","doi":"https://doi.org/10.5281/zenodo.23972681","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23972681","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23972681","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5159121649","display_name":"P. Graham","orcid":"https://orcid.org/0009-0007-8493-1505","observed_orcids":["https://orcid.org/0009-0007-8493-1505"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Graham, P.","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0007-8493-1505","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5159121649"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9955999851226807,"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":0.9955999851226807,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.0015999999595806003,"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":0.0013000000035390258,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/hales-jewett-theorem","display_name":"Hales-Jewett theorem","score":0.9710000157356262},{"id":"https://openalex.org/keywords/interval-coloring","display_name":"interval coloring","score":0.43700000643730164},{"id":"https://openalex.org/keywords/lower-bounds","display_name":"lower bounds","score":0.6869999766349792},{"id":"https://openalex.org/keywords/ramsey-theory","display_name":"Ramsey theory","score":0.6230000257492065},{"id":"https://openalex.org/keywords/lean-formalization","display_name":"Lean formalization","score":0.6460000276565552},{"id":"https://openalex.org/keywords/quadratic-lower-bound","display_name":"quadratic lower bound","score":0.43299999833106995}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8554999828338623},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.7444000244140625},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6855999827384949},{"id":"https://openalex.org/C195956108","wikidata":"https://www.wikidata.org/wiki/Q7268362","display_name":"Quadratic growth","level":2,"score":0.5486999750137329},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5302000045776367},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4657999873161316},{"id":"https://openalex.org/C40833965","wikidata":"https://www.wikidata.org/wiki/Q6901438","display_name":"Monochromatic color","level":2,"score":0.4602999985218048},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.4142000079154968},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.4092000126838684},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.34709998965263367},{"id":"https://openalex.org/C2777151079","wikidata":"https://www.wikidata.org/wiki/Q141160","display_name":"Parity (physics)","level":2,"score":0.32739999890327454},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.3264000117778778},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.2912999987602234},{"id":"https://openalex.org/C171988527","wikidata":"https://www.wikidata.org/wiki/Q188276","display_name":"Pigeonhole principle","level":2,"score":0.2531000077724457}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.5281/zenodo.23972681","doi":"https://doi.org/10.5281/zenodo.23972681","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23972681","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.23972681","doi":"https://doi.org/10.5281/zenodo.23972681","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23972681","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W2002353129","https://openalex.org/W2041110868","https://openalex.org/W2787194191","https://openalex.org/W2900207878","https://openalex.org/W4415381797","https://openalex.org/W7165636419","https://openalex.org/W7167209129"],"related_works":["https://openalex.org/W2900207878","https://openalex.org/W2787194191","https://openalex.org/W4415381797","https://openalex.org/W7165636419","https://openalex.org/W2460160336","https://openalex.org/W7225191892","https://openalex.org/W7167209129","https://openalex.org/W2994993691","https://openalex.org/W4403578587","https://openalex.org/W4393063801"],"abstract_inverted_index":{"A":[0],"combinatorial":[1],"line":[2,93,225],"in":[3,65,206,220,234,281],"$[m]^n$":[4,45,89],"is":[5,20,97,118,142,184,203,232,243],"$q$-fold":[6,49],"if":[7],"its":[8],"wildcard":[9,95],"set,":[10],"the":[11,18,34,116,143,148,156,160,177,185,188,195,207,224,237,239,260,264,277,284,296],"set":[12,96],"of":[13,23,28,44,74,88,100,125,155,179,187,191,194,217],"coordinates":[14],"that":[15,53,140],"vary":[16],"along":[17],"line,":[19,201],"a":[21,47,98,200,204,214,244],"union":[22,99],"at":[24,62,101,169],"most":[25,102,170],"$q$":[26,36],"intervals":[27],"consecutive":[29,192],"coordinates.":[30],"Let":[31],"$I(m,r)$":[32,242],"be":[33],"least":[35,63],"such":[37,199],"that,":[38],"for":[39,55,67,111],"$n$":[40],"large,":[41],"every":[42,56,68,112],"$r$-coloring":[43],"has":[46,251],"monochromatic":[48,92],"line.":[50],"We":[51],"show":[52],"$I(m,2)>(m-3)^2/128$":[54],"$m\\ge3$,":[57],"so":[58,223],"$I(m,r)\\ge":[59],"I(m,2)$":[60],"grows":[61],"quadratically":[64],"$m$":[66],"$r\\ge2$.":[69],"This":[70,229,249],"answers":[71],"Question":[72],"3.1":[73],"Conlon":[75,126],"and":[76,84,107,127,130,132,134,137,152,263,273],"Kam\u010dev,":[77,128],"who":[78],"asked":[79],"whether":[80],"some":[81],"$m\\ge4$,":[82],"$r\\ge2$":[83],"$c>1$":[85],"admit":[86],"$r$-colorings":[87],"with":[90,168],"no":[91],"whose":[94,210],"$cr$":[103],"intervals:":[104],"$m=18$,":[105],"$r=2$":[106],"any":[108],"$1<c<3/2$":[109],"do,":[110],"$n$.":[113],"For":[114],"$m=3$":[115],"value":[117],"known,":[119],"$r$":[120],"or":[121],"$r-1$,":[122],"by":[123,176,256,268,289,292],"work":[124,250],"Leader":[129],"R\u00e4ty,":[131],"Kam\u010dev":[133,136],"Spiegel;":[135],"Spiegel":[138],"noted":[139],"$r-1$":[141],"best":[144],"explicit":[145],"lower":[146],"bound":[147,162],"known":[149],"methods":[150],"give,":[151],"Shelah's":[153],"proof":[154],"Hales\u2013Jewett":[157],"theorem":[158,231],"gives":[159],"upper":[161],"$\\mathrm{HJ}(m-1,r)$.":[163],"To":[164],"rule":[165],"out":[166],"lines":[167],"$Q$":[171],"intervals,":[172],"we":[173],"color":[174],"$x$":[175],"parity":[178],"$\\lfloor":[180],"S(x)/(32Q)\\rfloor$,":[181],"where":[182],"$S(x)$":[183],"sum":[186],"squared":[189],"differences":[190],"letters":[193],"word":[196],"$0x$.":[197],"Along":[198],"$S$":[202],"quadratic":[205],"free":[208],"letter":[209],"values":[211],"rise":[212],"through":[213],"whole":[215],"interval":[216],"length":[218],"$32Q$":[219],"smaller":[221],"steps,":[222],"meets":[226],"both":[227],"colors.":[228],"coloring":[230],"proved":[233],"Lean":[235,261,282],"from":[236,283],"definitions;":[238,285],"step":[240],"to":[241],"short":[245],"argument":[246],"on":[247],"paper.":[248],"not":[252,286],"yet":[253,287],"been":[254],"reviewed":[255,288],"experts.":[257,290],"The":[258],"proofs,":[259],"formalization":[262],"text":[265],"were":[266],"produced":[267],"AI":[269,293],"models":[270,294],"(from":[271],"OpenAI":[272],"Anthropic),":[274],"working":[275],"under":[276,295],"author's":[278,297],"direction.":[279,298],"Checked":[280],"Produced":[291]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-11-06T00:00:00"}
