{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:45:23Z","timestamp":1787323523342,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100012652","name":"ETH Z\u00fcrich Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100012652","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DGE-1650112"],"award-info":[{"award-number":["DGE-1650112"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1800332"],"award-info":[{"award-number":["DMS-1800332"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1952786"],"award-info":[{"award-number":["DMS-1952786"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>For two graphs $F$ and $H$, the relative Tur\u00e1n number ${ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau [ SIAM J. Discrete Math., 29 (2015), pp. 65--78] and Perarnau and Reed [ Combin. Probab. Comput., 26 (2017), pp. 448--467] studied these quantities as a function of the maximum degree of $H$. In this paper, we study a generalization for uniform hypergraphs. If $F$ is a complete $r$-partite $r$-uniform hypergraph with parts of sizes $s_1,s_2,\\dots,s_r$ with each $s_{i + 1}$ sufficiently large relative to $s_i$, then with $1\/\\beta = \\sum_{i = 2}^r \\prod_{j = 1}^{i - 1} s_j$ we prove that for any $r$-uniform hypergraph $H$ with maximum degree $\\Delta$, ${ex}(H,F)\\ge \\Delta^{-\\beta - o(1)} \\cdot e(H).$ This is tight as $\\Delta \\rightarrow \\infty$ up to the $o(1)$ term in the exponent, since we show there exists a $\\Delta$-regular $r$-graph $H$ such that ${ex}(H,F)=O(\\Delta^{-\\beta}) \\cdot e(H)$. Similar tight results are obtained when $H$ is the random $n$-vertex $r$-graph $H_{n,p}^r$ with edge-probability $p$, extending results of Balogh and Samotij [ J. Lond. Math. Soc., 83 (2011), pp. 1091--1094] and Morris and Saxton [ Adv. Math., 298 (2016), pp. 534--580]. General lower bounds for a wider class of $F$ are also obtained.<\/jats:p>","DOI":"10.1137\/20m1364631","type":"journal-article","created":{"date-parts":[[2021,9,20]],"date-time":"2021-09-20T12:00:46Z","timestamp":1632139246000},"page":"2170-2191","source":"Crossref","is-referenced-by-count":3,"title":["Relative Tur\u00e1n Problems for Uniform Hypergraphs"],"prefix":"10.1137","volume":"35","author":[{"given":"Sam","family":"Spiro","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jacques","family":"Verstra\u00ebte","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,9,20]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1999.1906"},{"key":"atypb2","unstructured":"N. Alon and J. H. Spencer,\n                      The Probabilistic Method\n                      , John Wiley & Sons, Hoboken, NJ, 2004."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/jdq086"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1966-109-8"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.4171\/JEMS\/798"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.37236\/383"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"D. Conlon and W. T. Gowers,\n                      Combinatorial theorems in sparse random sets\n                      , Ann. Math. (2016), pp. 367-454.","DOI":"10.4007\/annals.2016.184.2.2"},{"key":"atypb8","unstructured":"J. Corsten and T. Tran,\n                      Balanced Supersaturation for Degenerate Hypergraphs\n                      , preprint, arXiv:1707.03788, 2017,https:\/\/arxiv.org\/abs\/1707.03788."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02759942"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9800(68)80051-1"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579292"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/140954416"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(94)90287-9"},{"key":"atypb14","first-page":"228","volume":"15","author":"Katona G.","year":"1964","journal-title":"Mat. Lapok"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"P. Keevash,\n                      Hypergraph Tur\u00e1n problems\n                      , in Surveys in Combinatorics 2011, Lond. Math. Soc. Lecture Note Ser. 392, Cambridge Univ. Press, Cambridge, UK, 2011, pp. 83-139.","DOI":"10.1017\/CBO9781139004114.004"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009804"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/BF01200906"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/BF01261323"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2017.10.002"},{"key":"atypb20","unstructured":"W. Mantel,\n                      Problem\n                      28, Wiskundige Opgaven, 10 (1907), pp. 60-61."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2016.05.001"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548301005028"},{"key":"atypb23","unstructured":"M. C. Neto and R. Morris,\n                      Maximum-Size Antichains in Random Set-Systems\n                      , preprint, arXiv:1404.5258, 2014,https:\/\/arxiv.org\/abs\/1404.5258."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548316000328"},{"key":"atypb25","unstructured":"C. Pohoata and D. Zakharov,\n                      Norm Hypergraphs\n                      , preprint, arXiv:2101.00715, 2021,https:\/\/arxiv.org\/abs\/2101.00715."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2016.184.2.1"},{"key":"atypb27","first-page":"436","volume":"48","author":"Tur\u00e1n P.","year":"1941","journal-title":"Mat. Fiz. Lapok"},{"key":"atypb28","first-page":"83","author":"Verstra\u00ebte J.","year":"2016","journal-title":"Cham"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(91)90097-4"},{"key":"atypb30","first-page":"19","volume":"2","author":"Zarankiewicz K.","year":"1954","journal-title":"Colloq. Math."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1364631","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:07:25Z","timestamp":1787321245000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1364631"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":30,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1364631"],"URL":"https:\/\/doi.org\/10.1137\/20m1364631","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}