On randomly generated intersecting hypergraphs II

TL;DR

This paper analyzes the structure of large random intersecting hypergraphs, revealing a typical configuration involving a core hypergraph, a special vertex, and all edges containing it, extending previous work to larger hyperedge sizes.

Contribution

It extends the understanding of the structure of random intersecting hypergraphs to larger hyperedge sizes, specifically for r growing faster than n^{1/3} but slower than n^{5/12}.

Findings

Whp hypergraph consists of a core, a distinguished vertex, and all edges containing it.

The size of the core hypergraph is Θ(r/n^{1/3}).

Results extend previous work to larger hyperedge sizes.

Abstract

Let $c$ be a positive constant. Suppose that $r=o(n^{5/12})$ and the members of $\binom{[n]}{r}$ are chosen sequentially at random to form an intersecting hypergraph $\mathcal{H}$. We show that whp $\mathcal{H}$ consists of a simple hypergraph $\mathcal{S}$ of size $\Theta(r/n^{1/3})$, a distinguished vertex $v$ and all $r$-sets which contain $v$ and meet every edge of $\mathcal{S}$. This is a continuation of the study of such random intersecting systems started in [Electron. J. Combin, (2003) R29] where the case $r=O(n^{1/3})$ was considered. To obtain the stated result we continue to investigate this question in the range $\omega(n^{1/3})\le r \le o(n^{5/12})$.