Supersaturation of odd linear cycles

Abstract

An $r$-uniform linear cycle of length $\ell$, denoted by $C_{\ell }^r$, 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 $0<\hat{I}´<1$ and $n$ sufficiently large, we show that every $n$-vertex $r$-graph $G$ with $n^{r-\hat{I}´}$ edges contains at least $n^{(r-1)(2\ell +1)-\hat{I}´(2\ell +1+\frac{4\ell -1}{(r-1)(2\ell +1)-3})-o(1)}$ copies of $C_{2\ell +1}^r$. Further, conditioning on the existence of dense high-girth hypergraphs, we show that there exists $n$-vertex $r$-graphs with $n^{r-\hat{I}´}$ edges and at most $n^{(r-1)(2\ell +1)-\hat{I}´(2\ell +1+\frac{1}{(r-1)\ell -1})+o(1)}$ copies of $C_{2\ell +1}^r$.

17 pages,1 figure