{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:15Z","timestamp":1753893855590,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A hypergraph is linear\u00a0if any two of its edges intersect in at most one vertex. The sail (or $3$-fan) $F^3$ is the $3$-uniform linear hypergraph consisting of $3$ edges $f_1, f_2, f_3$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting each $f_i$ in a vertex different from $v$. The linear Tur\u00e1n number $\\mathrm{ex}_{\\mathrm{lin}}(n, F^3)$ is the maximum number of edges in a $3$-uniform linear hypergraph on $n$ vertices that does not contain a copy of $F^3$.\r\nF\u00fcredi and Gy\u00e1rf\u00e1s proved that if $n = 3k$, then $\\mathrm{ex}_{\\mathrm{lin}}(n, F^3) = k^2$ and the only extremal hypergraphs in this case are transversal designs. They also showed that if $n = 3k+2$, then $\\mathrm{ex}_{\\mathrm{lin}}(n, F^3) = k^2+k$, and the only extremal hypergraphs are truncated designs (which are obtained from a transversal design on $3k+3$ vertices with $3$ groups by removing one vertex and all the hyperedges containing it) along with three other small hypergraphs. However, the case when $n =3k+1$ was left open.\r\nIn this paper, we solve this remaining case by proving that $\\mathrm{ex}_{\\mathrm{lin}}(n, F^3) = k^2+1$ if $n = 3k+1$, answering a question of F\u00fcredi and Gy\u00e1rf\u00e1s. We also characterize all the extremal hypergraphs. The difficulty of this case is due to the fact that these extremal examples are rather non-standard. In particular, they are not derived from transversal designs like in the other cases.<\/jats:p>","DOI":"10.37236\/9904","type":"journal-article","created":{"date-parts":[[2021,12,3]],"date-time":"2021-12-03T02:05:34Z","timestamp":1638497134000},"source":"Crossref","is-referenced-by-count":0,"title":["The Exact Linear Tur\u00e1n Number of the Sail"],"prefix":"10.37236","volume":"28","author":[{"given":"Beka","family":"Ergemlidze","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ervin","family":"Gy\u0151ri","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abhishek","family":"Methuku","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2021,12,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p39\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i4p39\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,3]],"date-time":"2021-12-03T02:05:35Z","timestamp":1638497135000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i4p39"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,3]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,10,8]]}},"URL":"https:\/\/doi.org\/10.37236\/9904","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2021,12,3]]},"article-number":"P4.39"}}