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

Intersections of hypergraphs

Given two weighted k-uniform hypergraphs G, H of order n, how much (or little) can we make them overlap by placing them on the same vertex set? If we place them at random, how concentrated is the distribution of the intersection? The aim of this paper is to investigate these questions

Exploring hypergraphs with martingales

Recently, we adapted exploration and martingale arguments of Nachmias and Peres, in turn based on ideas of Martin-L\"of, Karp and Aldous, to prove asymptotic normality of the number $L_1$ of vertices in the largest component $C$ of the random $r$-uniform hypergraph throughout the supercritical regime. In this paper we take these arguments further to prove two new results: strong tail bounds on the distribution of $L_1$, and joint asymptotic normality of $L_1$ and the number $M_1$ of edges of $C$. These results are used in a separate paper "Counting connected hypergraphs via the probabilistic method" to enumerate sparsely connected hypergraphs asymptotically.Comment: 32 pages; significantly expanded presentation. To appear in Random Structures and Algorithm

A note on the Harris-Kesten Theorem

Recently, a short proof of the Harris-Kesten result that the critical probability for bond percolation in the planar square lattice is 1/2 was given, using a sharp threshold result of Friedgut and Kalai. Here we point out that a key part of this proof may be replaced by an argument of Russo from 1982, using his approximate zero-one law in place of the Friedgut-Kalai result. Russo's paper gave a new proof of the Harris-Kesten Theorem that seems to have received little attention.Comment: 4 pages; author list changed, acknowledgement adde

Positive independence densities of finite rank countable hypergraphs are achieved by finite hypergraphs

The independence density of a finite hypergraph is the probability that a subset of vertices, chosen uniformly at random contains no hyperedges. Independence densities can be generalized to countable hypergraphs using limits. We show that, in fact, every positive independence density of a countably infinite hypergraph with hyperedges of bounded size is equal to the independence density of some finite hypergraph whose hyperedges are no larger than those in the infinite hypergraph. This answers a question of Bonato, Brown, Kemkes, and Pra{\l}at about independence densities of graphs. Furthermore, we show that for any $k$, the set of independence densities of hypergraphs with hyperedges of size at most $k$ is closed and contains no infinite increasing sequences.Comment: To appear in the European Journal of Combinatorics, 12 page

Hereditary properties of partitions, ordered graphs and ordered hypergraphs

In this paper we use the Klazar-Marcus-Tardos method to prove that if a hereditary property of partitions P has super-exponential speed, then for every k-permutation pi, P contains the partition of [2k] with parts {i, pi(i) + k}, where 1 <= i <= k. We also prove a similar jump, from exponential to factorial, in the possible speeds of monotone properties of ordered graphs, and of hereditary properties of ordered graphs not containing large complete, or complete bipartite ordered graphs. Our results generalize the Stanley-Wilf Conjecture on the number of n-permutations avoiding a fixed permutation, which was recently proved by the combined results of Klazar and of Marcus and Tardos. Our main results follow from a generalization to ordered hypergraphs of the theorem of Marcus and Tardos.Comment: 25 pgs, no figure

Packing random graphs and hypergraphs

We determine to within a constant factor the threshold for the property that two random k-uniform hypergraphs with edge probability p have an edge-disjoint packing into the same vertex set. More generally, we allow the hypergraphs to have different densities. In the graph case, we prove a stronger result, on packing a random graph with a fixed graph

Hereditary properties of combinatorial structures: posets and oriented graphs

A hereditary property of combinatorial structures is a collection of structures (e.g. graphs, posets) which is closed under isomorphism, closed under taking induced substructures (e.g. induced subgraphs), and contains arbitrarily large structures. Given a property P, we write P_n for the collection of distinct (i.e., non-isomorphic) structures in a property P with n vertices, and call the function n -> |P_n| the speed (or unlabelled speed) of P. Also, we write P^n for the collection of distinct labelled structures in P with vertices labelled 1,...,n, and call the function n -> |P^n| the labelled speed of P. The possible labelled speeds of a hereditary property of graphs have been extensively studied, and the aim of this paper is to investigate the possible speeds of other combinatorial structures, namely posets and oriented graphs. More precisely, we show that (for sufficiently large n), the labelled speed of a hereditary property of posets is either 1, or exactly a polynomial, or at least 2^n - 1. We also show that there is an initial jump in the possible unlabelled speeds of hereditary properties of posets, tournaments and directed graphs, from bounded to linear speed, and give a sharp lower bound on the possible linear speeds in each case.Comment: 26 pgs, no figure