{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T14:13:37Z","timestamp":1780668817981,"version":"3.54.1"},"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>An $r$-uniform linear cycle of length $\\ell$, denoted by $C^r_{\\ell}$, is an $r$-graph with $\\ell$ edges $e_1,e_2,\\dots,e_{\\ell}$ where $e_i=\\{v_{(r-1)(i-1)},v_{(r-1)(i-1)+1},\\dots,v_{(r-1)i}\\}$ (here $v_0=v_{(r-1)\\ell}$). For $0&lt;\\delta&lt;1$ and $n$ sufficiently large, we show that every $n$-vertex $r$-graph $G$ with $n^{r-\\delta}$ edges contains at least $n^{(r-1)(2\\ell+1)-\\delta(2\\ell+1+\\frac{4\\ell-1}{(r-1)(2\\ell+1)-3})-o(1)}$ copies of $C^r_{2\\ell+1}$. Further, conditioning on the existence of dense high-girth hypergraphs, we show that there exists $n$-vertex $r$-graphs with $n^{r-\\delta}$ edges and at most $n^{(r-1)(2\\ell+1)-\\delta(2\\ell+1+\\frac{1}{(r-1)\\ell-1})+o(1)}$ copies of $C^r_{2\\ell+1}$.<\/jats:p>","DOI":"10.37236\/14165","type":"journal-article","created":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T13:31:34Z","timestamp":1780666294000},"source":"Crossref","is-referenced-by-count":0,"title":["Supersaturation of Odd Linear Cycles"],"prefix":"10.37236","volume":"33","author":[{"given":"Lirong","family":"Deng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jie","family":"Han","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiaxi","family":"Nie","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sam","family":"Spiro","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[2026,6,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i2p49\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i2p49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T13:31:34Z","timestamp":1780666294000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i2p49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,5]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,4,14]]}},"URL":"https:\/\/doi.org\/10.37236\/14165","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,5]]},"article-number":"P2.49"}}