{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T14:54:01Z","timestamp":1773068041335,"version":"3.50.1"},"publisher-location":"New York, NY","reference-count":74,"publisher":"Springer New York","isbn-type":[{"value":"9781461472537","type":"print"},{"value":"9781461472544","type":"electronic"}],"license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2013]]},"DOI":"10.1007\/978-1-4614-7254-4_16","type":"book-chapter","created":{"date-parts":[[2013,8,1]],"date-time":"2013-08-01T18:35:43Z","timestamp":1375382143000},"page":"207-232","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Flag Algebras: An Interim Report"],"prefix":"10.1007","author":[{"given":"Alexander A.","family":"Razborov","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2013,5,18]]},"reference":[{"key":"16_CR1","unstructured":"R.\u00a0Baber. Some results in extremal combinatorics. PhD thesis, University College London, 2011."},{"issue":"1","key":"16_CR2","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1006\/jcta.2002.3285","volume":"100","author":"J. Balogh","year":"2002","unstructured":"J.\u00a0Balogh. The Turan density of triple systems is not principal. Journal of Combinatorial Theory, ser. A, 100(1):176\u2013180, 2002.","journal-title":"Journal of Combinatorial Theory, ser. A"},{"key":"16_CR3","doi-asserted-by":"crossref","unstructured":"I.\u00a0Barany. A generalization of Carath\u00e9odory\u2019s theorem. Discrete Mathematics, 40, 1982.","DOI":"10.1016\/0012-365X(82)90115-7"},{"key":"16_CR4","doi-asserted-by":"crossref","first-page":"227","DOI":"10.4064\/fm-69-3-227-231","volume":"69","author":"M. Behzad","year":"1970","unstructured":"M.\u00a0Behzad, G.\u00a0Chartrand, and C.\u00a0E. Wall. On minimal regular digraphs with given girth. Fundamenta Mathematicae, 69:227\u2013231, 1970.","journal-title":"Fundamenta Mathematicae"},{"key":"16_CR5","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/BF00181519","volume":"17","author":"E. Boros","year":"1984","unstructured":"E.\u00a0Boros and Z.\u00a0F\u00fcredi. The number of triangles covering the center of an n-set. Geom. Dedicata, 17:69\u201377, 1984.","journal-title":"Geom. Dedicata"},{"key":"16_CR6","unstructured":"J.\u00a0Balogh, P.\u00a0Hu, B.\u00a0Lidick\u00fd, and H.\u00a0Liu. Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube. Technical Report 1201.0209 [math.CO], arXiv, 2012."},{"key":"16_CR7","doi-asserted-by":"crossref","unstructured":"B.\u00a0Bollob\u00e1s. Relations between sets of complete subgraphs. In Proc. Fifth British Comb. Conference, pages\u00a079\u201384, 1975.","DOI":"10.1017\/S0305004100052063"},{"issue":"1","key":"16_CR8","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1017\/S0305004100052063","volume":"79","author":"B. Bollob\u00e1s","year":"1976","unstructured":"B.\u00a0Bollob\u00e1s. On complete subgraphs of different orders. Mathematical Proceedings of the Cambridge Philosophical Society, 79(1):19\u201324, 1976.","journal-title":"Mathematical Proceedings of the Cambridge Philosophical Society"},{"key":"16_CR9","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/S0012-365X(96)00162-8","volume":"165\/166","author":"J. A. Bondy","year":"1997","unstructured":"J.\u00a0A. Bondy. Counting subgraphs: A new approach to the Caccetta-H\u00e4ggkvist conjecture. Discrete Math., 165\/166:71\u201380, 1997.","journal-title":"Discrete Math."},{"issue":"1","key":"16_CR10","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1080\/10556789908805765","volume":"11","author":"B Borchers","year":"1999","unstructured":"B\u00a0Borchers. CSDP, a C library for semidefinite programming. Optimization Methods and Software, 11(1):613\u2013623, 1999.","journal-title":"Optimization Methods and Software"},{"key":"16_CR11","doi-asserted-by":"crossref","unstructured":"W.\u00a0G. Brown. On an open problem of Paul Tur\u00e1n concerning 3-graphs. In\u00a0Studies in pure mathematics, pages\u00a091\u201393. Birkh\u00e4user, 1983.","DOI":"10.1007\/978-3-0348-5438-2_9"},{"issue":"2","key":"16_CR12","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1017\/S0963548310000222","volume":"20","author":"R. Baber","year":"2011","unstructured":"R.\u00a0Baber and J.\u00a0Talbot. Hypergraphs do jump. Combinatorics, Probability and Computing, 20(2):161\u2013171, 2011.","journal-title":"Combinatorics, Probability and Computing"},{"issue":"2","key":"16_CR13","doi-asserted-by":"crossref","first-page":"P22","DOI":"10.37236\/2360","volume":"19","author":"R. Baber","year":"2012","unstructured":"R.\u00a0Baber and J.\u00a0Talbot. New Tur\u00e1n densities for 3-graphs. Electronic Journal of Combinatorics, 19(2):P22, 2012.","journal-title":"Electronic Journal of Combinatorics"},{"key":"16_CR14","unstructured":"D.\u00a0de Caen. The current status of Tur\u00e1n problem on hypergraphs. In\u00a0Extremal Problems for Finite Sets, Visegr\u00e1d (Hungary), volume\u00a03, pages 187\u2013197. Bolyai Society Mathematical Studies, 1991."},{"key":"16_CR15","doi-asserted-by":"crossref","DOI":"10.1201\/9781439863879","volume-title":"Erd\u0151s on Graphs: His Legacy of Unsolved Problems","author":"F. Chung","year":"1998","unstructured":"F.\u00a0Chung and R.\u00a0Graham. Erd\u0151s on Graphs: His Legacy of Unsolved Problems. A. K. Peters, 1998."},{"key":"16_CR16","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/BF02125347","volume":"9","author":"F. R. K. Chung","year":"1989","unstructured":"F.\u00a0R.\u00a0K. Chung, R.\u00a0L. Graham, and R.\u00a0M. Wilson. Quasi-random graphs. Combinatorica, 9:345\u2013362, 1989.","journal-title":"Combinatorica"},{"key":"16_CR17","first-page":"181","volume":"21","author":"L. Caccetta","year":"1978","unstructured":"L.\u00a0Caccetta and R.\u00a0H\u00e4ggkvist. On minimal digraphs with given girth. Congressus Numerantium, 21:181\u2013187, 1978.","journal-title":"Congressus Numerantium"},{"key":"16_CR18","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1006\/jcta.1998.2961","volume":"87","author":"F. Chung","year":"1999","unstructured":"F.\u00a0Chung and L.\u00a0Lu. An upper bound for the Tur\u00e1n number t\n                3(n,\u20094). Journal of Combinatorial Theory (A), 87:381\u2013389, 1999.","journal-title":"Journal of Combinatorial Theory (A)"},{"issue":"1","key":"16_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00493-008-2331-z","volume":"28","author":"M. Chudnovsky","year":"2008","unstructured":"M.\u00a0Chudnovsky, P.\u00a0Seymour, and B.\u00a0Sullivan. Cycles in dense graphs. Combinatorica, 28(1):1\u201318, 2008.","journal-title":"Combinatorica"},{"key":"16_CR20","unstructured":"S.\u00a0Das, H.\u00a0Huang, J.\u00a0Ma, H.\u00a0Naves, and B.\u00a0Sudakov. A problem of Erd\u0151s on the minimum number of k-cliques. Technical Report 1203.2723 [math.CO], arXiv, 2012."},{"key":"16_CR21","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1007\/s10107-006-0039-7","volume":"09","author":"E. de Klerk","year":"2007","unstructured":"E.\u00a0de\u00a0Klerk, D.\u00a0V. Pasechnik, and A.\u00a0Schrijver. Reduction of symmetric semidefinite programs using the regular $*$-representation. Mathematical programming, 09:613\u2013624, 2007.","journal-title":"Mathematical programming"},{"key":"16_CR22","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1002\/jgt.3190130607","volume":"13","author":"P. Erd\u0151s","year":"1989","unstructured":"P.\u00a0Erd\u0151s, R.\u00a0Faudree, A.\u00a0Gy\u00e1rf\u00e1s, and R.\u00a0H. Schelp. Domination in colored complete graphs. Journal of Graph Theory, 13:713\u2013718, 1989.","journal-title":"Journal of Graph Theory"},{"key":"16_CR23","first-page":"459","volume":"7","author":"P. Erd\u0151s","year":"1962","unstructured":"P.\u00a0Erd\u0151s. On the number of complete subgraphs contained in certain graphs. Publ. Math. Inst. Hungar. Acad. Sci, 7:459\u2013464, 1962.","journal-title":"Publ. Math. Inst. Hungar. Acad. Sci"},{"key":"16_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0012-365X(71)90002-1","volume":"1","author":"P. Erd\u0151s","year":"1971","unstructured":"P.\u00a0Erd\u0151s. On some extremal problems on r-graphs. Discrete Mathematics, 1:1\u20136, 1971.","journal-title":"Discrete Mathematics"},{"key":"16_CR25","unstructured":"P.\u00a0Erd\u0151s. On some problems in graph theory, combinatorial analysis and combinatorial number theory. In Graph theory and combinatorics (Cambridge 1983), pages\u00a01\u201317, 1984."},{"key":"16_CR26","doi-asserted-by":"publisher","first-page":"1087","DOI":"10.1090\/S0002-9904-1946-08715-7","volume":"52","author":"P. Erd\u0151s","year":"1946","unstructured":"P.\u00a0Erd\u0151s and A.\u00a0H. Stone. On the structure of linear graphs. Bulletin of the American Mathematical Society, 52:1087\u20131091, 1946.","journal-title":"Bulletin of the American Mathematical Society"},{"key":"16_CR27","first-page":"51","volume":"1","author":"P. Erd\u0151s","year":"1966","unstructured":"P.\u00a0Erd\u0151s and M.\u00a0Simonovits. A limit theorem in graph theory. Stud. Sci. Math. Hungar., 1:51\u201357, 1966.","journal-title":"Stud. Sci. Math. Hungar."},{"key":"16_CR28","doi-asserted-by":"crossref","unstructured":"D.\u00a0G. Fon-der Flaass. Method for construction of (3,4)-graphs. Mathematical Notes, 44(4):781\u2013783, 1988. Translated from Matematicheskie Zametki, Vol.\u00a044, No.\u00a04, pp.\u00a0546\u2013550, 1988.","DOI":"10.1007\/BF01158925"},{"key":"16_CR29","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1016\/0012-365X(84)90058-X","volume":"50","author":"P. Frankl","year":"1984","unstructured":"P.\u00a0Frankl and Z.\u00a0F\u00fcredi. An exact result for 3-graphs. Discrete Mathematics, 50:323\u2013328, 1984.","journal-title":"Discrete Mathematics"},{"issue":"4","key":"16_CR30","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1002\/jgt.3190130411","volume":"13","author":"D. Fisher","year":"1989","unstructured":"D.\u00a0Fisher. Lower bounds on the number of triangles in a graph. Journal of Graph Theory, 13(4):505\u2013512, 1989.","journal-title":"Journal of Graph Theory"},{"key":"16_CR31","doi-asserted-by":"crossref","unstructured":"Z.\u00a0F\u00fcredi, O.\u00a0Pikhurko, and M.\u00a0Simonovits. The Tur\u00e1n density of the hypergraph {abc,\u2009ade,\u2009bde.\u2009cde}. Electronic Journal of Combinatorics, R18, 2003.","DOI":"10.37236\/1711"},{"key":"16_CR32","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/BF02579215","volume":"4","author":"P. Frankl","year":"1984","unstructured":"P.\u00a0Frankl and V.\u00a0R\u00f6dl. Hypergraphs do not jump. Combinatorica, 4:149\u2013159, 1984.","journal-title":"Combinatorica"},{"issue":"3","key":"16_CR33","doi-asserted-by":"crossref","first-page":"P40","DOI":"10.37236\/2733","volume":"19","author":"V. Falgas-Ravry","year":"2012","unstructured":"V.\u00a0Falgas-Ravry and E.\u00a0R. Vaughan. Tur\u00e1n H-densities for 3-graphs. The\u00a0Electronic Journal of Combinatorics, 19(3):P40, 2012.","journal-title":"The Electronic Journal of Combinatorics"},{"key":"16_CR34","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1017\/S0963548312000508","volume":"22","author":"V. Falgas-Ravry","year":"2013","unstructured":"V.\u00a0Falgas-Ravry and E.\u00a0R. Vaughan. Applications of the semi-definite method to the Tur\u00e1n density problem for 3-graphs. Combinatorics, Probability and Computing, 22:21\u201354, 2013.","journal-title":"Combinatorics, Probability and Computing"},{"issue":"9","key":"16_CR35","doi-asserted-by":"publisher","first-page":"778","DOI":"10.2307\/2310464","volume":"66","author":"A. W. Goodman","year":"1959","unstructured":"A.\u00a0W. Goodman. On sets of acquaintances and strangers at any party. American Mathematical Monthly, 66(9):778\u2013783, 1959.","journal-title":"American Mathematical Monthly"},{"key":"16_CR36","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1007\/s00039-010-0073-8","volume":"20","author":"M. Gromov","year":"2010","unstructured":"M.\u00a0Gromov. Singularities, expanders and topology of maps. Part\u00a02: From combinatorics to topology via algebraic isoperimetry. Geometric and Functional Analysis, 20:416\u2013526, 2010.","journal-title":"Geometric and Functional Analysis"},{"key":"16_CR37","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.1016\/j.jctb.2012.04.001","volume":"102","author":"A. Grzesik","year":"2012","unstructured":"A.\u00a0Grzesik. On the maximum number of five-cycles in a triangle-free graph. Journal of Combinatorial Theory, ser. B, 102:1061\u20131066, 2012.","journal-title":"Journal of Combinatorial Theory, ser. B"},{"issue":"1","key":"16_CR38","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/BF02122689","volume":"9","author":"E. Gyori","year":"1989","unstructured":"E.\u00a0Gyori. On the number of C\n                5\u2019s in a triangle-free graph. Combinatorica, 9(1):101\u2013102, 1989.","journal-title":"Combinatorica"},{"key":"16_CR39","doi-asserted-by":"crossref","unstructured":"P.\u00a0Hamburger, P.\u00a0Haxell, and A.\u00a0Kostochka. On directed triangles in digraphs. Electronic Journal of Combinatorics, 14(1):Note 19, 2007.","DOI":"10.37236\/1020"},{"key":"16_CR40","unstructured":"Hatami H, J.\u00a0Hladky, D.\u00a0Kral, S.\u00a0Norin, and A.\u00a0Razborov. On the number of pentagons in triangle-free graphs. Technical Report 1102.1634v1 [math.CO], arXiv, 2011."},{"issue":"5","key":"16_CR41","doi-asserted-by":"crossref","first-page":"734","DOI":"10.1017\/S0963548312000107","volume":"21","author":"H Hatami","year":"2012","unstructured":"Hatami H, J.\u00a0Hladky, D.\u00a0Kral, S.\u00a0Norin, and A.\u00a0Razborov. Non-three-colorable common graphs exist. Combinatorics, Probability and Computing, 21(5):734\u2013742, 2012.","journal-title":"Combinatorics, Probability and Computing"},{"key":"16_CR42","unstructured":"J.\u00a0Hirst. The inducibility of graphs on four vertices. Technical Report 1109.1592 [math.CO], arXiv, 2011."},{"key":"16_CR43","doi-asserted-by":"crossref","unstructured":"J.\u00a0Hladk\u00fd, D.\u00a0Kr\u00e1l\u2019, and S.\u00a0Norin. Counting flags in triangle-free digraphs. Technical Report 0908.2791 [math.CO], arXiv, 2009.","DOI":"10.1016\/j.endm.2009.07.105"},{"key":"16_CR44","doi-asserted-by":"publisher","first-page":"547","DOI":"10.1090\/S0894-0347-2010-00687-X","volume":"24","author":"H. Hatami","year":"2011","unstructured":"H.\u00a0Hatami and S.\u00a0Norin. Undecidability of linear inequalities in graph homomorphism densities. Journal of the American Mathematical Society, 24:547\u2013565, 2011.","journal-title":"Journal of the American Mathematical Society"},{"key":"16_CR45","doi-asserted-by":"crossref","unstructured":"P.\u00a0Keevash. Hypergraph Tur\u00e1n problems. In R.\u00a0Chapman, editor, Surveys in Combinatorics, pages 83\u2013140. Cambridge University Press, 2011.","DOI":"10.1017\/CBO9781139004114.004"},{"key":"16_CR46","doi-asserted-by":"crossref","unstructured":"D.\u00a0Kr\u00e1l\u2019, C.\u00a0Liu, J.\u00a0Sereni, P.\u00a0Whalen, and Z.\u00a0Yilma. A new bound for the 2\/3 conjecture. Technical Report 1204.2519 [math.CO], arXiv, 2013.","DOI":"10.1017\/S0963548312000612"},{"key":"16_CR47","doi-asserted-by":"crossref","unstructured":"D.\u00a0Kr\u00e1l\u2019, L.\u00a0Mach, and J.\u00a0Sereni. A new lower bound based on Gromov\u2019s method of selecting heavily covered points. Technical Report 1108.0297 [math.CO], arXiv, 2012.","DOI":"10.1007\/s00454-012-9419-3"},{"key":"16_CR48","first-page":"344","volume":"4","author":"N. Khad\u017eiivanov","year":"1978","unstructured":"N.\u00a0Khad\u017eiivanov and V.\u00a0Nikiforov. The Nordhaus-Stewart-Moon-Moser inequality. Serdica, 4:344\u2013350, 1978. In Russian.","journal-title":"Serdica"},{"issue":"2","key":"16_CR49","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1007\/BF02579317","volume":"2","author":"A. V. Kostochka","year":"1982","unstructured":"A.\u00a0V. Kostochka. A class of constructions for Tur\u00e1n\u2019s (3, 4)-problem. Combinatorica, 2(2):187\u2013192, 1982.","journal-title":"Combinatorica"},{"key":"16_CR50","doi-asserted-by":"crossref","unstructured":"L.\u00a0Lov\u00e1sz. Large Networks and Graph Limits. American Mathematical Society, 2012.","DOI":"10.1090\/coll\/060"},{"key":"16_CR51","doi-asserted-by":"crossref","unstructured":"L.\u00a0Lov\u00e1sz and M.\u00a0Simonovits. On the number of complete subgraphs of a graph, II. In Studies in pure mathematics, pages\u00a0459\u2013495. Birkha\u00fcser, 1983.","DOI":"10.1007\/978-3-0348-5438-2_41"},{"key":"16_CR52","first-page":"283","volume":"7","author":"J. W. Moon","year":"1962","unstructured":"J.\u00a0W. Moon and L.\u00a0Moser. On a problem of Tur\u00e1n. Magyar. Tud. Akad. Mat. Kutat\u00f3 Int. K\u00f6zl, 7:283\u2013286, 1962.","journal-title":"Magyar. Tud. Akad. Mat. Kutat\u00f3 Int. K\u00f6zl"},{"issue":"19","key":"16_CR53","doi-asserted-by":"publisher","first-page":"4430","DOI":"10.1016\/j.disc.2007.08.040","volume":"308","author":"D. Mubayi","year":"2008","unstructured":"D.\u00a0Mubayi and O.\u00a0Pikhurko. Constructions of nonprincipal families in extremal hypergraph theory. Discrete Mathematics, 308(19):4430\u20134434, 2008.","journal-title":"Discrete Mathematics"},{"key":"16_CR54","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1006\/jcta.2002.3284","volume":"100","author":"D. Mubayi","year":"2002","unstructured":"D.\u00a0Mubayi and V.\u00a0R\u00f6dl. On the Tur\u00e1n number of triple systems. Journal of Combinatorial Theory, Ser. A, 100:136\u2013152, 2002.","journal-title":"Journal of Combinatorial Theory, Ser. A"},{"key":"16_CR55","doi-asserted-by":"publisher","first-page":"533","DOI":"10.4153\/CJM-1965-053-6","volume":"17","author":"T. S. Motzkin","year":"1965","unstructured":"T.\u00a0S. Motzkin and E.\u00a0G. Straus. Maxima for graphs and a new proof of a theorem of Tur\u00e1n. Canadian Journal of Mathematics, 17:533\u2013540, 1965.","journal-title":"Canadian Journal of Mathematics"},{"key":"16_CR56","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1017\/S0963548301004722","volume":"10","author":"V. Nikiforov","year":"2001","unstructured":"V.\u00a0Nikiforov. On the minimum number of k-cliques in graphs with restricted independence number. Combinatorics, Probability and Computing, 10:361\u2013366, 2001.","journal-title":"Combinatorics, Probability and Computing"},{"issue":"3","key":"16_CR57","doi-asserted-by":"publisher","first-page":"1599","DOI":"10.1090\/S0002-9947-2010-05189-X","volume":"363","author":"V. Nikiforov","year":"2011","unstructured":"V.\u00a0Nikiforov. The number of cliques in graphs of given order and size. Transactions of the American Mathematical Society, 363(3):1599\u20131618, 2011.","journal-title":"Transactions of the American Mathematical Society"},{"key":"16_CR58","doi-asserted-by":"publisher","first-page":"33","DOI":"10.4153\/CJM-1963-004-7","volume":"15","author":"E. A. Nordhaus","year":"1963","unstructured":"E.\u00a0A. Nordhaus and B.\u00a0M. Stewart. Triangles in an ordinary graph. Canadian J. Math., 15:33\u201341, 1963.","journal-title":"Canadian J. Math."},{"key":"16_CR59","doi-asserted-by":"publisher","first-page":"1142","DOI":"10.1016\/j.ejc.2011.03.006","volume":"23","author":"O. Pikhurko","year":"2011","unstructured":"O.\u00a0Pikhurko. The minimum size of 3-graphs without a 4-set spanning no or exactly three edges. European Journal of Combinatorics, 23:1142\u20131155, 2011.","journal-title":"European Journal of Combinatorics"},{"key":"16_CR60","unstructured":"O.\u00a0Pikhurko. Minimum number of k-cliques in graphs with bounded independence number. Technical Report 1204.4423 [math.CO], arXiv, 2012."},{"key":"16_CR61","unstructured":"O.\u00a0Pikhurko. On possible Tur\u00e1n densities. Technical Report 1204.4423 [math.CO], arXiv, 2012."},{"key":"16_CR62","unstructured":"O.\u00a0Pikhurko and A.\u00a0Razborov. Asymptotic structure of graphs with the minimum number of triangles. Technical Report 1204.2846v1 [math.CO], arXiv, 2012."},{"issue":"4","key":"16_CR63","doi-asserted-by":"publisher","first-page":"1239","DOI":"10.2178\/jsl\/1203350785","volume":"72","author":"A. Razborov","year":"2007","unstructured":"A.\u00a0Razborov. Flag algebras. Journal of Symbolic Logic, 72(4):1239\u20131282, 2007.","journal-title":"Journal of Symbolic Logic"},{"issue":"4","key":"16_CR64","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1017\/S0963548308009085","volume":"17","author":"A. Razborov","year":"2008","unstructured":"A.\u00a0Razborov. On the minimal density of triangles in graphs. Combinatorics, Probability and Computing, 17(4):603\u2013618, 2008.","journal-title":"Combinatorics, Probability and Computing"},{"issue":"3","key":"16_CR65","doi-asserted-by":"publisher","first-page":"946","DOI":"10.1137\/090747476","volume":"24","author":"A. Razborov","year":"2010","unstructured":"A.\u00a0Razborov. On 3-hypergraphs with forbidden 4-vertex configurations. SIAM Journal on Discrete Mathematics, 24(3):946\u2013963, 2010.","journal-title":"SIAM Journal on Discrete Mathematics"},{"key":"16_CR66","unstructured":"A.\u00a0Razborov. On the Caccetta-Haggkvist conjecture with forbidden subgraphs. Technical Report 1107.2247v1 [math.CO], arXiv, 2011."},{"key":"16_CR67","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1134\/S0081543811060150","volume":"274","author":"A. Razborov","year":"2011","unstructured":"A.\u00a0Razborov. On the Fon-der-Flaass interpretation of extremal examples for Turan\u2019s (3,4)-problem. Proceedings of the Steklov Institute of Mathematics, 274:247\u2013266, 2011.","journal-title":"Proceedings of the Steklov Institute of Mathematics"},{"key":"16_CR68","unstructured":"A.\u00a0Razborov. On Turan\u2019s (3,4)-problem with forbidden configurations. Technical Report 1210.4605v1 [math.CO], arXiv, 2012."},{"key":"16_CR69","unstructured":"C.\u00a0Reiher. The clique density theorem. Technical Report 1212.2454 [math.CO], arXiv, 2012."},{"issue":"2","key":"16_CR70","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1006\/jctb.1998.1839","volume":"74","author":"J. Shen","year":"1998","unstructured":"J.\u00a0Shen. Directed triangles in graphs. Journal of Combinatorial Theory Ser. B, 74(2):405\u2013407, 1998.","journal-title":"Journal of Combinatorial Theory Ser. B"},{"key":"16_CR71","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/BF01929486","volume":"11","author":"A. F. Sidorenko","year":"1995","unstructured":"A.\u00a0F. Sidorenko. What we know and what we do not know about Tur\u00e1n numbers. Graphs and Combinatorics, 11:179\u2013199, 1995.","journal-title":"Graphs and Combinatorics"},{"key":"16_CR72","unstructured":"K.\u00a0Sperfeld. The inducibility of small oriented graphs. Technical Report 1111.4813 [math.CO], arXiv, 2011."},{"key":"16_CR73","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1112\/jlms\/s2-39.2.246","volume":"39","author":"A. Thomason","year":"1989","unstructured":"A.\u00a0Thomason. A disproof of a conjecture of Erd\u0151s in Ramsey theory. Journal of the London Mathematical Society, 39:246\u2013255, 1989.","journal-title":"Journal of the London Mathematical Society"},{"key":"16_CR74","first-page":"436","volume":"48","author":"P. Tur\u00e1n","year":"1941","unstructured":"P.\u00a0Tur\u00e1n. Egy gr\u00e1felm\u00e9leti sz\u00e9ls\u00f6\u00e9rt\u00e9kfeladatr\u00f3l. Mat. \u00e9s Fiz. Lapok, 48:436\u2013453, 1941.","journal-title":"Mat. \u00e9s Fiz. Lapok"}],"container-title":["The Mathematics of Paul Erd\u0151s II"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-1-4614-7254-4_16","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T14:54:26Z","timestamp":1720709666000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-1-4614-7254-4_16"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"ISBN":["9781461472537","9781461472544"],"references-count":74,"URL":"https:\/\/doi.org\/10.1007\/978-1-4614-7254-4_16","relation":{},"subject":[],"published":{"date-parts":[[2013]]},"assertion":[{"value":"18 May 2013","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}