{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T21:06:24Z","timestamp":1775163984929,"version":"3.50.1"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":2293,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2007,12]]},"abstract":"Abstract<\/jats:title>Asymptotic extremal combinatorics deals with questions that in the language of model theory can be re-stated as follows. For finite models M, N<\/jats:italic> of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p<\/jats:italic>(M, N<\/jats:italic>) be the probability that a randomly chosen sub-model of N<\/jats:italic> with \u2223M\u2223 elements is isomorphic to M<\/jats:italic>. Which asymptotic relations exist between the quantities p<\/jats:italic>(M<\/jats:italic>1<\/jats:sub>,N<\/jats:italic>),\u2026, p<\/jats:italic>(M<\/jats:italic>h<\/jats:sub>,N<\/jats:italic>), where M<\/jats:italic>1<\/jats:sub>,\u2026, M<\/jats:italic>1<\/jats:sub>, are fixed \u201ctemplate\u201d models and \u2223N\u2223 grows to infinity?<\/jats:p>In this paper we develop a formal calculus that captures many standard arguments in the area, both previously known and apparently new. We give the first application of this formalism by presenting a new simple proof of a result by Fisher about the minimal possible density of triangles in a graph with given edge density.<\/jats:p>","DOI":"10.2178\/jsl\/1203350785","type":"journal-article","created":{"date-parts":[[2008,3,25]],"date-time":"2008-03-25T14:42:11Z","timestamp":1206456131000},"page":"1239-1282","source":"Crossref","is-referenced-by-count":171,"title":["Flag algebras"],"prefix":"10.1017","volume":"72","author":[{"given":"Alexander A.","family":"Razborov","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200004904_ref002","first-page":"79","volume-title":"Proceedings of the Fifth British Combinatorial Conference","author":"Bollob\u00e1s","year":"1976"},{"key":"S0022481200004904_ref009","volume-title":"Probabilistic methods in combinatorics","author":"Erd\u00f6s","year":"1974"},{"key":"S0022481200004904_ref017","unstructured":"Lov\u00e1sz L. and Szegedy B. , Limits of dense graph sequences, Technical Report TR-2004-79, Microsoft Research, 08 2004."},{"key":"S0022481200004904_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579292"},{"key":"S0022481200004904_ref015","first-page":"419","volume-title":"Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science","author":"Grigoriev","year":"2002"},{"key":"S0022481200004904_ref011","unstructured":"Felix D. , Asymptotic relations in flag algebras, Manuscript, 2005."},{"key":"S0022481200004904_ref019","unstructured":"Podolski V. 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