Asymptotic enumeration of linear hypergraphs with given number of vertices and edges

For $n\ge 3$, let $r=r(n)\ge 3$ be an integer. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear $r$-uniform hypergraphs on $n\to \mathrm{\infty }$ vertices is determined asymptotically when the number of edges is $m(n)=o(r^{-3}{n}^{\frac{3}{2}})$. As one application, we find the probability of linearity for the independent-edge model of random $r$-uniform hypergraph when the expected number of edges is $o(r^{-3}n^{ \frac32})$\mathrm{.}Wealsofindtheprobabilitythatarandom$r$-uniform linear hypergraph with a given number of edges contains a given subhypergraph.