{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T09:09:36Z","timestamp":1763543376308,"version":"3.45.0"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2025,8,11]],"date-time":"2025-08-11T00:00:00Z","timestamp":1754870400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We consider the hypergraph Tur\u00e1n problem of determining\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100096_inline1.png\"\/>\n                        <jats:tex-math>$ex(n, S^d)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , the maximum number of facets in a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100096_inline2.png\"\/>\n                        <jats:tex-math>$d$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -dimensional simplicial complex on\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100096_inline3.png\"\/>\n                        <jats:tex-math>$n$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    vertices that does not contain a simplicial\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100096_inline4.png\"\/>\n                        <jats:tex-math>$d$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -sphere (a\n                    <jats:italic>homeomorph<\/jats:italic>\n                    of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100096_inline5.png\"\/>\n                        <jats:tex-math>$S^d$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ) as a subcomplex. We show that if there is an affirmative answer to a question of Gromov about sphere enumeration in high dimensions, then\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100096_inline6.png\"\/>\n                        <jats:tex-math>$ex(n, S^d) \\geq \\Omega (n^{d + 1 - (d + 1)\/(2^{d + 1} - 2)})$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . Furthermore, this lower bound holds unconditionally for 2-LC (locally constructible) spheres, which includes all shellable spheres and therefore all polytopes. We also prove an upper bound on\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100096_inline7.png\"\/>\n                        <jats:tex-math>$ex(n, S^d)$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100096_inline8.png\"\/>\n                        <jats:tex-math>$O(n^{d + 1 - 1\/2^{d - 1}})$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    using a simple induction argument. We conjecture that the upper bound can be improved to match the conditional lower bound.\n                  <\/jats:p>","DOI":"10.1017\/s0963548325100096","type":"journal-article","created":{"date-parts":[[2025,8,11]],"date-time":"2025-08-11T05:43:33Z","timestamp":1754891013000},"page":"848-856","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["A conditional lower bound for the Tur\u00e1n number of spheres"],"prefix":"10.1017","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4666-0218","authenticated-orcid":false,"given":"Andrew","family":"Newman","sequence":"first","affiliation":[{"name":"Carnegie Mellon University"}]},{"given":"Marta","family":"Pavelka","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University"}]}],"member":"56","published-online":{"date-parts":[[2025,8,11]]},"reference":[{"key":"S0963548325100096_ref15","first-page":"12","article-title":"Simplicial homeomorphs and trace-bounded hypergraphs","volume":"6","author":"Long","year":"2022","journal-title":"Discrete Anal."},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref2","DOI":"10.4171\/aihpd\/170"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref24","DOI":"10.4153\/CJM-1962-002-9"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref13","DOI":"10.1007\/s004540010043"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref1","DOI":"10.1090\/S0002-9947-2012-05614-5"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref23","DOI":"10.1007\/BF02018585"},{"unstructured":"[7] Gromov, M. (2000) Spaces and questions. Geom. Funct. Anal. 118\u2013161. GAFA. 2000 (Tel Aviv, 1999).","key":"S0963548325100096_ref7"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref12","DOI":"10.1093\/imrn\/rnab099"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref5","DOI":"10.1016\/0550-3213(95)00207-9"},{"unstructured":"[14] Linial, N. (2018) Challenges of high-dimensional combinatorics. In Lov\u00e1sz\u2019s Seventieth Birthday Conference.","key":"S0963548325100096_ref14"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref9","DOI":"10.1007\/BF02187893"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref16","DOI":"10.1007\/BF01364272"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref6","DOI":"10.1007\/s11512-009-0119-z"},{"key":"S0963548325100096_ref25","volume-title":"Graduate Texts in Mathematics","volume":"152","author":"Ziegler","year":"1995"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref17","DOI":"10.1007\/s00208-015-1232-x"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref11","DOI":"10.1007\/s11856-021-2156-7"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref20","DOI":"10.1063\/1.533333"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref10","DOI":"10.1017\/CBO9781139004114.004"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref19","DOI":"10.1007\/BF02733251"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref4","DOI":"10.7146\/math.scand.a-11045"},{"key":"S0963548325100096_ref18","first-page":"1","volume-title":"Spectroscopic and Group Theoretical Methods in Physics","author":"Ponzano","year":"1968"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref3","DOI":"10.1007\/s11511-011-0062-2"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref8","DOI":"10.1007\/s00209-017-1981-1"},{"unstructured":"[21] Sankar, M. An improved Tur\u00e1n exponent for 2-complexes. arXiv:2408.09029.","key":"S0963548325100096_ref21"},{"doi-asserted-by":"publisher","key":"S0963548325100096_ref22","DOI":"10.1112\/blms.13167"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548325100096","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T09:04:46Z","timestamp":1763543086000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548325100096\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,11]]},"references-count":25,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,11]]}},"alternative-id":["S0963548325100096"],"URL":"https:\/\/doi.org\/10.1017\/s0963548325100096","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"type":"print","value":"0963-5483"},{"type":"electronic","value":"1469-2163"}],"subject":[],"published":{"date-parts":[[2025,8,11]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}