On randomly generated intersecting hypergraphs

TL;DR

This paper analyzes the properties of randomly generated intersecting hypergraphs, showing that under certain conditions, they tend to be of maximum size with a specific probability as the number of elements grows large.

Contribution

It provides a probabilistic analysis of the maximum size of intersecting hypergraphs generated randomly, revealing a limiting probability related to the parameter c.

Findings

Probability of maximum size approaches (1+c^3)^{-1} as n increases.

Maximum size hypergraphs are achieved with high probability under the given random process.

The size of the hypergraph is approximately ${n-1race r-1}$ for large n.

Abstract

Let $c$ be a positive constant. We show that if $r=\lfloor cn^{1/3}\rfloor$ and the members of ${[n]\choose r}$ are chosen sequentially at random to form an intersecting hypergraph then with limiting probability $(1+c^3)^{-1}$, as $n\to\infty$, the resulting family will be of maximum size ${n-1\choose r-1}$.