{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T00:28:25Z","timestamp":1769732905058,"version":"3.49.0"},"reference-count":30,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2020,6,23]],"date-time":"2020-06-23T00:00:00Z","timestamp":1592870400000},"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":[[2020,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A classical result of Erd\u0151s and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an <jats:italic>n<\/jats:italic>-vertex graph not containing <jats:italic>C<jats:sub>2k<\/jats:sub><\/jats:italic>, the cycle of length 2<jats:italic>k<\/jats:italic>, is <jats:italic>O<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>1+1\/<jats:italic>k<\/jats:italic><\/jats:sup>). Simonovits established a corresponding supersaturation result for <jats:italic>C<\/jats:italic><jats:sub>2<jats:italic>k<\/jats:italic><\/jats:sub>\u2019s, showing that there exist positive constants <jats:italic>C<\/jats:italic>,<jats:italic>c<\/jats:italic> depending only on <jats:italic>k<\/jats:italic> such that every <jats:italic>n<\/jats:italic>-vertex graph <jats:italic>G<\/jats:italic> with <jats:italic>e<\/jats:italic>(<jats:italic>G<\/jats:italic>)\u2a7e <jats:italic>Cn<\/jats:italic><jats:sup>1+1\/<jats:italic>k<\/jats:italic><\/jats:sup> contains at least <jats:italic>c<\/jats:italic>(<jats:italic>e<\/jats:italic>(<jats:italic>G<\/jats:italic>)\/<jats:italic>v<\/jats:italic>(<jats:italic>G<\/jats:italic>))<jats:sup>2<jats:italic>k<\/jats:italic><\/jats:sup> copies of <jats:italic>C<\/jats:italic><jats:sub>2<jats:italic>k<\/jats:italic><\/jats:sub>, this number of copies tightly achieved by the random graph (up to a multiplicative constant).<\/jats:p><jats:p>In this paper we extend Simonovits' result to a supersaturation result of <jats:italic>r<\/jats:italic>-uniform linear cycles of even length in <jats:italic>r<\/jats:italic>-uniform linear hypergraphs. Our proof is self-contained and includes the <jats:italic>r<\/jats:italic> = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.<\/jats:p>","DOI":"10.1017\/s0963548320000206","type":"journal-article","created":{"date-parts":[[2020,6,23]],"date-time":"2020-06-23T08:01:21Z","timestamp":1592899281000},"page":"698-721","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":5,"title":["Supersaturation of even linear cycles in linear hypergraphs"],"prefix":"10.1017","volume":"29","author":[{"given":"Tao","family":"Jiang","sequence":"first","affiliation":[]},{"given":"Liana","family":"Yepremyan","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,6,23]]},"reference":[{"key":"S0963548320000206_ref28","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(72)90028-3"},{"key":"S0963548320000206_ref27","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300004478"},{"key":"S0963548320000206_ref26","unstructured":"[26] Simonovits, M. (1982) Extremal graph problems, degenerate extremal problems, and supersaturated graphs. In Progress in Graph Theory (Waterloo 1982), Academic Press, pp. 419\u2013437."},{"key":"S0963548320000206_ref24","first-page":"489","article-title":"Inequalities for functionals generated by bipartite graphs (in Russian). Diskret. Mat. 3 50\u201365. English translation","volume":"2","author":"Sidorenko","year":"1991","journal-title":"Discrete Math. Appl."},{"key":"S0963548320000206_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2016.05.001"},{"key":"S0963548320000206_ref6","first-page":"51","article-title":"A limit theorem in graph theory","volume":"1","author":"Erd\u0151s","year":"1966","journal-title":"Studia Sci. Math. Hungar."},{"key":"S0963548320000206_ref10","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1946-08715-7"},{"key":"S0963548320000206_ref22","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(85)80023-8"},{"key":"S0963548320000206_ref18","unstructured":"[18] Keevash, P. (2014) The existence of designs. arXiv:1401.3665."},{"key":"S0963548320000206_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2014.09.005"},{"key":"S0963548320000206_ref15","first-page":"169","volume-title":"Studies","author":"F\u00fcredi","year":"2013"},{"key":"S0963548320000206_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2013.12.009"},{"key":"S0963548320000206_ref30","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(75)90067-9"},{"key":"S0963548320000206_ref2","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/BF02018585","article-title":"On the existence of triangulated spheres in 3-graphs and related problems","volume":"3","author":"Brown","year":"1973","journal-title":"Period. Math. Hungar."},{"key":"S0963548320000206_ref23","unstructured":"[23] Ruzsa, I. and Szemer\u00e9di, E. Triples systems with no six points carrying three triangles. In Combinatorics II (Keszthely 1976), Vol. 18 of Colloquia Mathematica Societatis J\u00e1nos Bolyai, pp. 939\u2013945."},{"key":"S0963548320000206_ref7","unstructured":"[7] Erd\u0151s, P. and Simonovits, M. (1970) Some extremal problems in graph theory. In Combinatorial Theory and its Applications I (Proc. Colloq. Balatonf\u00fcred, 1969), North-Holland, pp. 377\u2013390."},{"key":"S0963548320000206_ref9","unstructured":"[9] Erd\u0151s, P. and Simonovits, M. (1984) Cube-supersaturated graphs and related problems. In Progress in Graph Theory (Waterloo, Ont., 1982), Academic Press, pp. 203\u2013218."},{"key":"S0963548320000206_ref16","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032718"},{"key":"S0963548320000206_ref29","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(72)90029-5"},{"key":"S0963548320000206_ref25","doi-asserted-by":"publisher","DOI":"10.1007\/BF02988307"},{"key":"S0963548320000206_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2018.07.003"},{"key":"S0963548320000206_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(74)90052-5"},{"key":"S0963548320000206_ref4","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548316000134"},{"key":"S0963548320000206_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548317000530"},{"key":"S0963548320000206_ref21","unstructured":"[21] Pikhurko, O. (2012) A note on the Tur\u00e1n function of even cycles. Proc. Amer. Math. Soc. 140 3687\u20133692."},{"key":"S0963548320000206_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579292"},{"key":"S0963548320000206_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2018.12.004"},{"key":"S0963548320000206_ref17","first-page":"83","volume-title":"Surveys in Combinatorics","author":"Keevash","year":"2011"},{"key":"S0963548320000206_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579343"},{"key":"S0963548320000206_ref13","unstructured":"[13] Faudree, R. and Simonovits, M. Cycle-supersaturated graphs. In preparation."}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548320000206","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,14]],"date-time":"2020-09-14T12:06:10Z","timestamp":1600085170000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548320000206\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,23]]},"references-count":30,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["S0963548320000206"],"URL":"https:\/\/doi.org\/10.1017\/s0963548320000206","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,23]]},"assertion":[{"value":"\u00a9 The Author(s), 2020. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}