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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.05116 |
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| _version_ | 1866917981109354496 |
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| author | Deng, Lirong Han, Jie Nie, Jiaxi Spiro, Sam |
| author_facet | Deng, Lirong Han, Jie Nie, Jiaxi Spiro, Sam |
| contents | 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 $0<δ<1$ and $n$ sufficiently large, we show that every $n$-vertex $r$-graph $G$ with $n^{r-δ}$ edges contains at least $n^{(r-1)(2\ell+1)-δ(2\ell+1+\frac{4\ell-1}{(r-1)(2\ell+1)-3})-o(1)}$ 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 $n^{r-δ}$ edges and at most $n^{(r-1)(2\ell+1)-δ(2\ell+1+\frac{1}{(r-1)\ell-1})+o(1)}$ copies of $C^r_{2\ell+1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Supersaturation of odd linear cycles Deng, Lirong Han, Jie Nie, Jiaxi Spiro, Sam Combinatorics 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 $0<δ<1$ and $n$ sufficiently large, we show that every $n$-vertex $r$-graph $G$ with $n^{r-δ}$ edges contains at least $n^{(r-1)(2\ell+1)-δ(2\ell+1+\frac{4\ell-1}{(r-1)(2\ell+1)-3})-o(1)}$ 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 $n^{r-δ}$ edges and at most $n^{(r-1)(2\ell+1)-δ(2\ell+1+\frac{1}{(r-1)\ell-1})+o(1)}$ copies of $C^r_{2\ell+1}$. |
| title | Supersaturation of odd linear cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2504.05116 |