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 with m edges, whenever and . We give an asymptotic formula for the number of connected r‐uniform hypergraphs on with m edges, whenever is fixed and with , that is, the average degree tends to infinity. This complements recent results of Behrisch, Coja‐Oghlan, and Kang (the case ) and the present authors (the case , ie, “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.