{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T14:53:42Z","timestamp":1773068022448,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>If $\\mathcal{F}$ is a family of graphs then the Tur\u00e1n density of $\\mathcal{F}$ is determined by the minimum chromatic number of the members of $\\mathcal{F}$.The situation for Tur\u00e1n densities of 3-graphs is far more complex and still very unclear. Our aim in this paper is to present new exact Tur\u00e1n densities for individual and finite families of $3$-graphs, in many cases we are also able to give corresponding stability results. As well as providing new examples of individual $3$-graphs with Tur\u00e1n densities equal to $2\/9,4\/9,5\/9$ and $3\/4$ we also\u00a0give examples of irrational Tur\u00e1n densities for finite families of 3-graphs, disproving a conjecture of Chung and Graham. (Pikhurko has independently disproved this conjecture by a very different method.)A central question in this area, known as Tur\u00e1n's problem, is to determine the Tur\u00e1n density of $K_4^{(3)}=\\{123, 124, 134, 234\\}$. Tur\u00e1n conjectured that this should be $5\/9$. Razborov [On 3-hypergraphs with forbidden 4-vertex configurations in SIAM J. Disc. Math. 24 (2010), 946--963] showed\u00a0that if we consider the induced Tur\u00e1n problem forbidding $K_4^{(3)}$ and $E_1$, the 3-graph with 4 vertices and a single edge, then the Tur\u00e1n density is indeed $5\/9$. We give some new non-induced results of a similar nature, in particular we show that $\\pi(K_4^{(3)},H)=5\/9$ for a $3$-graph $H$ satisfying $\\pi(H)=3\/4$.We end with a number of open questions focusing mainly on the topic of which values can occur as Tur\u00e1n densities.Our work is mainly computational, making use of Razborov's flag algebra framework. However all proofs are exact in the sense that they can be verified without the use of any floating point operations. Indeed all verifying computations use only integer operations, working either over $\\mathbb{Q}$ or in the case of irrational Tur\u00e1n densities over an appropriate quadratic extension of $\\mathbb{Q}$.<\/jats:p>","DOI":"10.37236\/2360","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:28:54Z","timestamp":1578713334000},"source":"Crossref","is-referenced-by-count":21,"title":["New Tur\u00e1n Densities for 3-Graphs"],"prefix":"10.37236","volume":"19","author":[{"given":"Rahil","family":"Baber","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John","family":"Talbot","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2012,5,31]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i2p22\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i2p22\/4205","content-type":"application\/zip","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i2p22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:35:53Z","timestamp":1579300553000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i2p22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,5,31]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2012,4,7]]}},"URL":"https:\/\/doi.org\/10.37236\/2360","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,5,31]]},"article-number":"P22"}}