On the number of linear hypergraphs of large girth

An $r$-uniform \textit{linear cycle} of length $\ell$, denoted by $C_{\ell}^r$, is an $r$-graph with edges $e_1, \ldots, e_{\ell}$ such that for every $i\in [\ell-1]$, $|e_i\cap e_{i+1}|=1$, $|e_{\ell}\cap e_1|=1$ and $e_i\cap e_j=\emptyset$ for all other pairs $\{i, j\},\ i\neq j$. For every $r\geq 3$ and $\ell\geq 4$, we show that there exists a constant $C$ depending on $r$ and $\ell$ such that the number of linear $r$-graphs of girth $\ell$ is at most $2^{Cn^{1+1/\lfloor \ell/2\rfloor}}$. Furthermore, we extend the result for $\ell=4$, proving that there exists a constant $C$ depending on $r$ such that the number of linear $r$-graphs without $C_{4}^r$ is at most $2^{Cn^{3/2}}$. The idea of the proof is to reduce the hypergraph enumeration problems to some graph enumeration problems, and then apply a variant of the graph container method, which may be of independent interest. We extend a breakthrough result of Kleitman and Winston on the number of $C_4$-free graphs, proving that the number of graphs containing at most $n^2/32\log^6 n$ $C_4$'s is at most $2^{11n^{3/2}}$, for sufficiently large $n$. We further show that for every $r\geq 3$ and $\ell\geq 2$, the number of graphs such that each of its edges is contained in only $O(1)$ cycles of length at most $2\ell$, is bounded by $2^{3(\ell+1)n^{1+1/\ell}}$ asymptotically.