On Erd\H{o}s-Ko-Rado for random hypergraphs I

TL;DR

This paper investigates the conditions under which a random hypergraph exhibits the Erd ext{o}s-Ko-Rado property, providing precise probabilistic thresholds for large hypergraphs with certain parameters.

Contribution

It offers a detailed characterization of the probability thresholds for the Erd ext{o}s-Ko-Rado property in random hypergraphs for specific ranges of parameters.

Findings

Identifies thresholds for p(n,k) where hypergraphs are likely EKR

Provides asymptotic probability results for large n

Focuses on hypergraphs with k<√(cn log n) for c<1/4

Abstract

A family of sets is intersecting if no two of its members are disjoint, and has the Erd\H{o}s-Ko-Rado property (or is EKR) if each of its largest intersecting subfamilies has nonempty intersection. Denote by $\mathcal{H}_k(n,p)$ the random family in which each $k$-subset of $\{1\dots n\}$ is present with probability $p$, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks: \[ \mbox{for what $p=p(n,k)$ is $\mathcal{H}_k(n,p)$ likely to be EKR?} \] Here, for fixed $c<1/4$, and $k< \sqrt{cn\log n}$ we give a precise answer to this question, characterizing those sequences $p=p(n,k)$ for which $$ \Pr(\mathcal{H}_k(n,p) \textrm{ is EKR}) \rightarrow 1 \textrm{ as } n\rightarrow \infty. $$