# Counting dense connected hypergraphs via the probabilistic method

## Abstract

In 1990 Bender, Canfield, and McKay gave an asymptotic formula for the number of connected graphs on [n] = {1, 2, . . . , n} with m edges, whenever n→∞ and (Formula presented.). We give an asymptotic formula for the number Cr(n,m) of connected r-uniform hypergraphs on [n] with m edges, whenever r ≥ 3 is fixed and m = m(n) with m/n→∞, that is, the average degree tends to infinity. This complements recent results of Behrisch, Coja-Oghlan, and Kang (the case m = n/(r − 1) + Θ(n)) and the present authors (the case m = n/(r − 1) + o(n), 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.

## Publication Title

Random Structures and Algorithms

## Recommended Citation

Bollobás, B., & Riordan, O.
(2018). Counting dense connected hypergraphs via the probabilistic method.* Random Structures and Algorithms**, 53* (2), 185-220.
https://doi.org/10.1002/rsa.20762