Counting dense connected hypergraphs via the probabilistic method

In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on $[n]=\{1,2,\ldots,n\}$ with $m$ edges, whenever $n\to\infty$ and $n-1\le m=m(n)\le \binom{n}{2}$. We give an asymptotic formula for the number $C_r(n,m)$ of connected $r$-uniform hypergraphs on $[n]$ with $m$ edges, whenever $r\ge 3$ is fixed and $m=m(n)$ with $m/n\to\infty$, i.e., the average degree tends to infinity. This complements recent results of Behrisch, Coja-Oghlan and Kang (the case $m=n/(r-1)+\Theta(n)$) and the present authors (the case $m=n/(r-1)+o(n)$, i.e., `nullity' or `excess' $o(n)$). The proof is based on probabilistic methods, and in particular on a bivariate local limit theorem for the number of vertices and edges in the largest component of a certain random hypergraph. The arguments are much simpler than in the sparse case; in particular, we can use `smoothing' techniques to directly prove the local limit theorem, without needing to first prove a central limit theorem.