{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T09:28:19Z","timestamp":1771406899068,"version":"3.50.1"},"reference-count":27,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T00:00:00Z","timestamp":1765929600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In the 1980s, Erd\u0151s and S\u00f3s initiated the study of Tur\u00e1n problems with a uniformity condition on the distribution of edges: the uniform Tur\u00e1n density of a hypergraph\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100308_inline1.png\"\/>\n                        <jats:tex-math>$H$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is the infimum over all\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100308_inline2.png\"\/>\n                        <jats:tex-math>$d$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100308_inline3.png\"\/>\n                        <jats:tex-math>$d$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    contains\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100308_inline4.png\"\/>\n                        <jats:tex-math>$H$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . In particular, they asked to determine the uniform Tur\u00e1n densities 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=\"S0963548325100308_inline5.png\"\/>\n                        <jats:tex-math>$K_4^{(3)-}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100308_inline6.png\"\/>\n                        <jats:tex-math>$K_4^{(3)}$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . After more than 30 years, the former was solved in [Israel J. Math. 211 (2016), 349 \u2013 366] and [J. Eur. Math. Soc. 20 (2018), 1139 \u2013 1159], while the latter still remains open. Till today, there are known constructions 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=\"S0963548325100308_inline7.png\"\/>\n                        <jats:tex-math>$3$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -uniform hypergraphs with uniform Tur\u00e1n density equal to\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100308_inline8.png\"\/>\n                        <jats:tex-math>$0$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100308_inline9.png\"\/>\n                        <jats:tex-math>$1\/27$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100308_inline10.png\"\/>\n                        <jats:tex-math>$4\/27$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100308_inline11.png\"\/>\n                        <jats:tex-math>$1\/4$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    only. We extend this list by a fifth value: we prove an easy to verify sufficient condition for the uniform Tur\u00e1n density to be equal to\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548325100308_inline12.png\"\/>\n                        <jats:tex-math>$8\/27$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and identify hypergraphs satisfying this condition.\n                  <\/jats:p>","DOI":"10.1017\/s0963548325100308","type":"journal-article","created":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T06:06:16Z","timestamp":1765951576000},"page":"280-289","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Hypergraphs with uniform Tur\u00e1n density equal to 8\/27"],"prefix":"10.1017","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4435-1534","authenticated-orcid":false,"given":"Frederik","family":"Garbe","sequence":"first","affiliation":[{"name":"Institute of Computer Science"},{"name":"Masaryk University"}]},{"given":"Daniel","family":"Il\u2019kovi\u010d","sequence":"additional","affiliation":[{"name":"Masaryk University"},{"name":"Leipzig University"},{"name":"University of Warwick"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8680-0890","authenticated-orcid":false,"given":"Daniel","family":"Kr\u00e1\u013e","sequence":"additional","affiliation":[{"name":"Masaryk University"},{"name":"Leipzig University"},{"name":"Max Planck Institute for Mathematics in the Sciences"}]},{"given":"Filip","family":"Ku\u010der\u00e1k","sequence":"additional","affiliation":[{"name":"Masaryk University"},{"name":"Max Planck Institute for Mathematics in the Sciences"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1582-5171","authenticated-orcid":false,"given":"Ander","family":"Lamaison","sequence":"additional","affiliation":[{"name":"Masaryk University"},{"name":"Institute for Basic Science (IBS)"}]}],"member":"56","published-online":{"date-parts":[[2025,12,17]]},"reference":[{"key":"S0963548325100308_ref26","doi-asserted-by":"publisher","DOI":"10.1007\/BF01929486"},{"key":"S0963548325100308_ref16","first-page":"60","article-title":"Problem 28","volume":"10","author":"Mantel","year":"1907","journal-title":"Wiskundige Opgaven"},{"key":"S0963548325100308_ref1","doi-asserted-by":"publisher","DOI":"10.1090\/tran\/8873"},{"key":"S0963548325100308_ref14","unstructured":"[14] Lamaison, A. 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