Counting connected hypergraphs via the probabilistic method

In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on $[n]$ with $m$ edges, whenever $n$ and the nullity $m-n+1$ tend to infinity. Asymptotic formulae for the number of connected $r$-uniform hypergraphs on $[n]$ with $m$ edges and so nullity $t=(r-1)m-n+1$ were proved by Karo\'nski and \L uczak for the case $t=o(\log n/\log\log n)$, and Behrisch, Coja-Oghlan and Kang for $t=\Theta(n)$. Here we prove such a formula for any $r\ge 3$ fixed, and any $t=t(n)$ satisfying $t=o(n)$ and $t\to\infty$ as $n\to\infty$. This leaves open only the (much simpler) case $t/n\to\infty$, which we will consider in future work. ( arXiv:1511.04739 ) Our approach is probabilistic. Let $H^r_{n,p}$ denote the random $r$-uniform hypergraph on $[n]$ in which each edge is present independently with probability $p$. Let $L_1$ and $M_1$ be the numbers of vertices and edges in the largest component of $H^r_{n,p}$. We prove a local limit theorem giving an asymptotic formula for the probability that $L_1$ and $M_1$ take any given pair of values within the `typical' range, for any $p=p(n)$ in the supercritical regime, i.e., when $p=p(n)=(1+\epsilon(n))(r-2)!n^{-r+1}$ where $\epsilon^3n\to\infty$ and $\epsilon\to 0$; our enumerative result then follows easily. Taking as a starting point the recent joint central limit theorem for $L_1$ and $M_1$, we use smoothing techniques to show that `nearby' pairs of values arise with about the same probability, leading to the local limit theorem. Behrisch et al used similar ideas in a very different way, that does not seem to work in our setting. Independently, Sato and Wormald have recently proved the special case $r=3$, with an additional restriction on $t$. They use complementary, more enumerative methods, which seem to have a more limited scope, but to give additional information when they do work.Comment: Expanded; asymptotics clarified - no significant mathematical changes. 67 pages (including appendix

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, . . . , n} with m edges, whenever n → ∞ and n − 1 ≤ m = m(n) ≤ (  n2). 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 → ∞, i.e., 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), 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.</p