Asymptotic linearity of binomial random hypergraphs via cluster expansion under graph-dependence

TL;DR

This paper investigates the probability that a binomial random hypergraph is linear, using cluster expansion techniques to derive more precise asymptotics and improve previous results, especially for the case when r=3.

Contribution

It introduces a cluster expansion approach to analyze the linearity probability of random hypergraphs, providing sharper asymptotic estimates than prior work.

Findings

Derived more precise asymptotics for linearity probability

Improved previous asymptotic bounds for r=3 and p=o(n^{-7/5})

Enhanced understanding of hypergraph linearity under graph dependence

Abstract

Let integer $n \ge 3$ and integer $r = r(n) \ge 3$. Define the binomial random $r$-uniform hypergraph $H_r(n, p)$ to be the $r$-uniform graph on the vertex set $[n]$ such that each $r$-set is an edge independently with probability $p$. A hypergraph is linear if every pair of hyperedges intersects in at most one vertex. We study the probability of linearity of random hypergraphs $H_r(n, p)$ via cluster expansion and give more precise asymptotics of the probability in question, improving the asymptotic probability of linearity obtained by McKay and Tian, in particular, when $r=3$ and $p = o(n^{-7/5})$.