{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T12:02:48Z","timestamp":1768219368285,"version":"3.49.0"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T00:00:00Z","timestamp":1759795200000},"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":[[2026,1]]},"abstract":"Abstract<\/jats:title>\n \n Let\n \n \n \n $K^r_n$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be the complete\n \n \n \n $r$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -uniform hypergraph on\n \n \n \n $n$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n vertices, that is, the hypergraph whose vertex set is\n \n \n \n $[n] \\, :\\! = \\{1,2,\\ldots ,n\\}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and whose edge set is\n \n \n \n $\\binom {[n]}{r}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We form\n \n \n \n $G^r(n,p)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n by retaining each edge of\n \n \n \n $K^r_n$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n independently with probability\n \n \n \n $p$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . An\n \n \n \n $r$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -uniform hypergraph\n \n \n \n $H\\subseteq G$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is\n \n \n \n $F$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -\n saturated<\/jats:italic>\n if\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n does not contain any copy of\n \n \n \n $F$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , but any missing edge of\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in\n \n \n \n $G$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n creates a copy of\n \n \n \n $F$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Furthermore, we say that\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is\n weakly<\/jats:italic>\n \n \n \n $F$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -\n saturated<\/jats:italic>\n in\n \n \n \n $G$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n if\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n does not contain any copy of\n \n \n \n $F$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , but the missing edges of\n \n \n \n $H$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in\n \n \n \n $G$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n can be added back one-by-one, in some order, such that every edge creates a new copy of\n \n \n \n $F$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . The smallest number of edges in an\n \n \n \n $F$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -saturated hypergraph in\n \n \n \n $G$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is denoted by\n \n \n \n ${\\textit {sat}}(G,F)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and in a weakly\n \n \n \n $F$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -saturated hypergraph in\n \n \n \n $G$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n by\n \n \n \n $\\mathop {\\mbox{$w$-${sat}$}}\\! (G,F)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In 2017, Kor\u00e1ndi and Sudakov initiated the study of saturation in random graphs, showing that for constant\n \n \n \n $p$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , with high probability\n \n \n \n ${\\textit {sat}}(G(n,p),K_s)=(1+o(1))n\\log _{\\frac {1}{1-p}}n$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and\n \n \n \n $\\mathop {\\mbox{$w$-${sat}$}}\\! (G(n,p),K_s)=\\mathop {\\mbox{$w$-${sat}$}}\\! (K_n,K_s)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Generalising their results, in this paper, we solve the saturation problem for random hypergraphs\n \n \n \n $G^r(n,p)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for cliques\n \n \n \n $K_s^r$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , for every\n \n \n \n $2\\le r \\lt s$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and constant\n \n \n \n $p$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1017\/s0963548325100229","type":"journal-article","created":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T07:09:06Z","timestamp":1759820946000},"page":"40-58","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Saturation in random hypergraphs"],"prefix":"10.1017","volume":"35","author":[{"given":"Sahar","family":"Diskin","sequence":"first","affiliation":[{"name":"Tel Aviv University"}]},{"given":"Ilay","family":"Hoshen","sequence":"additional","affiliation":[{"name":"Tel Aviv University"}]},{"given":"D\u00e1niel","family":"Kor\u00e1ndi","sequence":"additional","affiliation":[]},{"given":"Benny","family":"Sudakov","sequence":"additional","affiliation":[{"name":"ETH"}]},{"given":"Maksim","family":"Zhukovskii","sequence":"additional","affiliation":[{"name":"The University of Sheffield"}]}],"member":"56","published-online":{"date-parts":[[2025,10,7]]},"reference":[{"key":"S0963548325100229_ref5","doi-asserted-by":"publisher","DOI":"10.1137\/21M1456479"},{"key":"S0963548325100229_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejc.2023.103777"},{"key":"S0963548325100229_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2017.09.026"},{"key":"S0963548325100229_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/BF02582930"},{"key":"S0963548325100229_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2023.113572"},{"key":"S0963548325100229_ref19","first-page":"163","article-title":"On some properties of linear complexes","volume":"24","author":"Zykov","year":"1949","journal-title":"Mat. Sbornik N.S."},{"key":"S0963548325100229_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01904851"},{"key":"S0963548325100229_ref11","first-page":"189","volume-title":"Convexity and Graph Theory (Jerusalem, 1981), Volume 87 of North-Holland Math. Stud.","author":"Kalai","year":"1984"},{"key":"S0963548325100229_ref7","unstructured":"[7] Diskin, S. , Hoshen, I. , Kor\u00e1ndi, D. , Sudakov, B. and Zhukovskii, M. (2024) Saturation in random hypergraphs. arXiv preprint arXiv:2405.03061."},{"key":"S0963548325100229_ref4","first-page":"25","volume-title":"Beitr\u00e4ge zur Graphentheorie (Kolloquium, Manebach, 1967)","author":"Bollob\u00e1s","year":"1968"},{"key":"S0963548325100229_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(82)80025-5"},{"key":"S0963548325100229_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(85)90048-2"},{"key":"S0963548325100229_ref9","doi-asserted-by":"publisher","DOI":"10.2307\/2311408"},{"key":"S0963548325100229_ref13","unstructured":"[13] Kalinichenko, O. , Miralaei, M. , Mohammadian, A. and Tayfeh-Rezaie, B. (2023) Weak saturation numbers in random graphs. arXiv preprint arXiv:2306.10375."},{"key":"S0963548325100229_ref8","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.70009"},{"key":"S0963548325100229_ref2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.23079"},{"key":"S0963548325100229_ref15","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20703"},{"key":"S0963548325100229_ref16","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240070204"},{"key":"S0963548325100229_ref17","first-page":"45","volume-title":"Combinatorial surveys (Proc. Sixth British Combinatorial Conf., Royal Holloway Coll., Egham, 1977)","author":"Lov\u00e1sz","year":"1977"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548325100229","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T08:49:23Z","timestamp":1768207763000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548325100229\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,7]]},"references-count":19,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["S0963548325100229"],"URL":"https:\/\/doi.org\/10.1017\/s0963548325100229","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,7]]},"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"}]}}