{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T13:38:49Z","timestamp":1775828329548,"version":"3.50.1"},"reference-count":12,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2003,7,17]],"date-time":"2003-07-17T00:00:00Z","timestamp":1058400000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Random Struct Algorithms"],"published-print":{"date-parts":[[2003,10]]},"abstract":"Abstract<\/jats:title>Szemer\u00e9di's Regularity Lemma proved to be a powerful tool in the area of extremal graph theory J. Koml\u00f3s and M. Simonovits, Szemer\u00e9di's Regularity Lemma and its applications in graph theory, Combinatorics 2 (1996), 295\u2013352<\/jats:ext-link>. Many of its applications are based on the following technical fact: If G<\/jats:italic> is a k<\/jats:italic>\u2010partite graph with V<\/jats:italic>(G<\/jats:italic>) = \u222aV<\/jats:italic>i<\/jats:italic><\/jats:sub>, |V<\/jats:italic>i<\/jats:italic><\/jats:sub>| = n<\/jats:italic> for all i<\/jats:italic> \u2208 [k<\/jats:italic>], and all pairs {V<\/jats:italic>i<\/jats:italic><\/jats:sub>, V<\/jats:italic>j<\/jats:italic><\/jats:sub>}, 1 \u2264 i < j \u2264 k<\/jats:italic>, are \u03f5\u2010regular of density d<\/jats:italic>, then G<\/jats:italic> contains d<\/jats:italic>n<\/jats:italic>k<\/jats:italic><\/jats:sup>(1 + f<\/jats:italic>(\u03f5)) cliques K<\/jats:italic>, where f<\/jats:italic>(\u03f5) \u2192 0 as \u03f5 \u2192 0. The aim of this paper is to establish the analogous statement for 3\u2010uniform hypergraphs. Our result, to which we refer as The Counting Lemma, together with Theorem 3.5 of P. Frankl and V. R\u00f6dl [Extremal problems on set systems, Random Structures Algorithms 20(2) (2002), 131\u2013164<\/jats:ext-link>, a Regularity Lemma for Hypergraphs, can be applied in various situations as Szemer\u00e9di's Regularity Lemma is for graphs. Some of these applications are discussed in previous papers, as well as in upcoming papers, of the authors and others. \u00a9 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 264\u2013332, 2003<\/jats:p>","DOI":"10.1002\/rsa.10094","type":"journal-article","created":{"date-parts":[[2003,8,25]],"date-time":"2003-08-25T13:15:00Z","timestamp":1061817300000},"page":"264-332","source":"Crossref","is-referenced-by-count":24,"title":["Regularity properties for triple systems"],"prefix":"10.1002","volume":"23","author":[{"given":"Brendan","family":"Nagle","sequence":"first","affiliation":[]},{"given":"Vojt\u011bch","family":"R\u00f6dl","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2003,7,17]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01788085"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10017"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10075"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548302005503"},{"key":"e_1_2_1_6_2","first-page":"295","article-title":"Szemer\u00e9di's Regularity Lemma and its applications in graph theory","volume":"2","author":"Koml\u00f3s J.","year":"1996","journal-title":"Combinatorics"},{"key":"e_1_2_1_7_2","unstructured":"B.Nagle Regularity properties for triple systems Ph.D. Thesis Emory University Atlanta GA 1999."},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(00)00280-6"},{"key":"e_1_2_1_9_2","unstructured":"Y.Peng V.R\u00f6dl andJ.Skokan Counting small cliques in 3\u2010uniform hypergraphs Combin Probab Comput to appear."},{"key":"e_1_2_1_10_2","unstructured":"V.R\u00f6dlandJ.Skokan Uniformity of set systems submitted."},{"key":"e_1_2_1_11_2","series-title":"Combinatorics","first-page":"939","author":"Ruzsa I. Z.","year":"1978"},{"key":"e_1_2_1_12_2","first-page":"399","article-title":"Regular partitions of graphs","author":"Szemer\u00e9di E.","year":"1976","journal-title":"Proc Colloq Int CNRS"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.4064\/aa-27-1-199-245"}],"container-title":["Random Structures & Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Frsa.10094","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/rsa.10094","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,17]],"date-time":"2023-10-17T19:49:57Z","timestamp":1697572197000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/rsa.10094"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,7,17]]},"references-count":12,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2003,10]]}},"alternative-id":["10.1002\/rsa.10094"],"URL":"https:\/\/doi.org\/10.1002\/rsa.10094","archive":["Portico"],"relation":{},"ISSN":["1042-9832","1098-2418"],"issn-type":[{"value":"1042-9832","type":"print"},{"value":"1098-2418","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,7,17]]}}}