Counting 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} withmedges, whenevernand the nullitym−n+1 tend to infinity. LetCr(n,t) be the number of connectedr-uniform hypergraphs on [n] with nullityt= (r−1)m−n+1, wheremis the number of edges. Forr≥ 3, asymptotic formulae forCr(n,t) are known only for partial ranges of the parameters: in 1997 Karoński and Łuczak gave one fort=o(logn/log logn), and recently Behrisch, Coja-Oghlan and Kang gave one fort=Θ(n). Here we prove such a formula for any fixedr≥ 3 and anyt=t(n) satisfyingt=o(n) andt→∞ asn→∞, complementing the last result. This leaves open only the caset/n→∞, which we expect to be much simpler, and will consider in future work. 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. We deduce this from the corresponding central limit theorem by smoothing techniques.