Asymptotic enumeration of linear hypergraphs with given number of vertices and edges

TL;DR

This paper provides an asymptotic enumeration of linear r-uniform hypergraphs with a given number of vertices and edges, and explores probabilities related to linearity and subhypergraph containment.

Contribution

It determines the asymptotic count of linear r-uniform hypergraphs under certain edge constraints and analyzes related probabilistic properties.

Findings

Asymptotic enumeration formula for linear hypergraphs with specified vertices and edges.

Probability estimates for linearity in random hypergraph models.

Likelihood of subhypergraph containment in linear hypergraphs.

Abstract

For $n\geq 3$, let $r=r(n)\geq 3$ be an integer. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear $r$-uniform hypergraphs on $n\to\infty$ vertices is determined asymptotically when the number of edges is $m(n)=o(r^{-3}n^{ \frac32})$. As one application, we find the probability of linearity for the independent-edge model of random $r$-uniform hypergraph when the expected number of edges is $o(r^{-3}n^{ \frac32})$. We also find the probability that a random $r$-uniform linear hypergraph with a given number of edges contains a given subhypergraph.