Erd\H{o}s-Ko-Rado for random hypergraphs: asymptotics and stability

TL;DR

This paper studies the size and stability of the largest intersecting subhypergraphs in random hypergraphs, extending the Erdős-Ko-Rado theorem to probabilistic settings across different regimes of hypergraph parameters.

Contribution

It provides asymptotic results and stability conditions for the maximum intersecting families in random hypergraphs, covering all ranges of parameters $p$ and $k$, and improves understanding of probabilistic combinatorics.

Findings

Largest intersecting subhypergraph size approximates $p rac{k}{n} N$ for large $p$ when $k= heta(n)$.

Stability results hold for certain ranges of $p$ and $k$, indicating typical structure.

Behavior differs for $k=o(n)$, with precise size estimates in that regime.

Abstract

We investigate the asymptotic version of the Erd\H{o}s-Ko-Rado theorem for the random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$. For $2 \leq k(n) \leq n/2$, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability tending to 1 as $n\to\infty$, the largest intersecting subhypergraph of $\mathcal{H}^k(n,p)$ has size $(1+o(1))p\frac kn N$, for any $p\gg \frac nk\ln^2\!\left(\frac nk\right)D^{-1}$. This lower bound on $p$ is asymptotically best possible for $k=\Theta(n)$. For this range of $k$ and $p$, we are able to show stability as well. A different behavior occurs when $k = o(n)$. In this case, the lower bound on $p$ is almost optimal. Further, for the small interval $D^{-1}\ll p \leq (n/k)^{1-\varepsilon}D^{-1}$, the largest intersecting subhypergraph of $\mathcal{H}^k(n,p)$ has size $\Theta(\ln (pD)N D^{-1})$, provided that $k \gg \sqrt{n \ln n}$. Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in $\mathcal{H}^k(n,p)$, for essentially all values of $p$ and $k$.