{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T16:56:24Z","timestamp":1787504184422,"version":"build-2736575974"},"reference-count":31,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2008,11,4]],"date-time":"2008-11-04T00:00:00Z","timestamp":1225756800000},"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":[[2009,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We describe a method that we believe may be foundational for a comprehensive theory of generalized Tur\u00e1n problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that is, any 3\u2010graph on\n                    <jats:italic>n<\/jats:italic>\n                    vertices for which every pair of vertices is contained in more than\n                    <jats:italic>n<\/jats:italic>\n                    \/2 edges must contain a Fano plane, for\n                    <jats:italic>n<\/jats:italic>\n                    sufficiently large. For projective planes over fields of odd size\n                    <jats:italic>q<\/jats:italic>\n                    we show that the codegree threshold is between\n                    <jats:italic>n<\/jats:italic>\n                    \/2 \u2212\n                    <jats:italic>q<\/jats:italic>\n                    + 1 and\n                    <jats:italic>n<\/jats:italic>\n                    \/2, but for\n                    <jats:italic>PG<\/jats:italic>\n                    <jats:sub>2<\/jats:sub>\n                    (4) we find the somewhat surprising phenomenon that the threshold is less than (1\/2 \u2212 \u03f5)\n                    <jats:italic>n<\/jats:italic>\n                    for some small \u03f5 &gt; 0. We conclude by setting out a program for future developments of this method to tackle other problems. \u00a9 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009\n                  <\/jats:p>","DOI":"10.1002\/rsa.20249","type":"journal-article","created":{"date-parts":[[2008,11,4]],"date-time":"2008-11-04T05:58:44Z","timestamp":1225778324000},"page":"123-164","source":"Crossref","is-referenced-by-count":14,"title":["A hypergraph regularity method for generalized Tur\u00e1n problems"],"prefix":"10.1002","volume":"34","author":[{"given":"Peter","family":"Keevash","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2008,11,4]]},"reference":[{"key":"e_1_2_1_2_2","first-page":"43","volume-title":"Blocking sets in the projective plane of order four, In Annals of Discrete Mathematics","author":"Berardi L.","year":"1988"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240010108"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02125347"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1999.1938"},{"key":"e_1_2_1_6_2","unstructured":"A.DiwanandCh. 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