{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:37Z","timestamp":1753893817786,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>An $(n,k,t)$-graph is a graph on $n$ vertices in which every set of $k$ vertices contains a clique on $t$ vertices. Tur\u00e1n's Theorem, rephrased in terms of graph complements, states that the unique minimum $(n,k,2)$-graph is an equitable disjoint union of cliques. We prove that minimum $(n,k,t)$-graphs are always disjoint unions of cliques for any $t$ (despite nonuniqueness of extremal examples), thereby generalizing Tur\u00e1n's Theorem and confirming two conjectures of Hoffman et al.<\/jats:p>","DOI":"10.37236\/12707","type":"journal-article","created":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T13:51:54Z","timestamp":1739541114000},"source":"Crossref","is-referenced-by-count":0,"title":["A Proof of the $(n,k,t)$ Conjectures"],"prefix":"10.37236","volume":"32","author":[{"given":"Stacie","family":"Baumann","sequence":"first","affiliation":[]},{"given":"Joseph","family":"Briggs","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2025,2,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i1p22\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i1p22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T13:51:55Z","timestamp":1739541115000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i1p22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1,17]]}},"URL":"https:\/\/doi.org\/10.37236\/12707","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,2,14]]},"article-number":"P1.22"}}