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For $n\geq 3$ and $r=r(n) \geq 3$, let $\boldsymbol{k}=\boldsymbol{k}(n)=(k_1, \ldots, k_n)$ be a sequence of non-negative integerswith sum $M(\boldsymbol{k})=\sum_{j=1}^{n} k_j$. We assume that$M(\boldsymbol{k})$ is divisible by $r$ for infinitely many values of $n$, andrestrict our attention to these values. Let $X=X(n)$ be a simple $r$-uniformhypergraph on the vertex set $V=\{v_1,v_2, \ldots, v_n\}$ with $t$ edges andmaximum degree $x_{\max}$. We denote by $\mathcal{H}_r(\boldsymbol{k})$ the setof all simple $r$-uniform hypergraphs on the vertex set $V$ with degreesequence $\boldsymbol{k}$, and let $\mathcal{H}_r(\boldsymbol{k},X)$ be the setof all hypergraphs in $\mathcal{H}_r(\boldsymbol{k})$ which contain no edge of$X$. We give an asymptotic enumeration formula for the size of$\mathcal{H}_r(\boldsymbol{k},X)$. This formula holds when $r^4k_{\max}^3=o(M(\boldsymbol{k}))$, $t\, k_{\max}^{3}\, =o(M(\boldsymbol{k})^2)$and $r\,t\,k_{\max}^4 = o(M(\boldsymbol{k})^3)$. Our proof involves theswitching method.  As a corollary, we obtain an asymptotic formula for the number of hypergraphsin $\mathcal{H}_r(\boldsymbol{k})$ which contain every edge of $X$. We applythis result to find asymptotic expressions for the expected number of perfectmatchings and loose Hamilton cycles in a random hypergraph in$\mathcal{H}_r(\boldsymbol{k})$ in the regular case.

Enumerating sparse uniform hypergraphs with given degree sequence and forbidden edges